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The Hyperbolic Universe

Salah. A . Mabkhout

Dhamar University ϭ

Ϯ

To my parents

ϯ

ϰ

Preface Cosmology, the study of the universe at its largest scales, is undergoing a period of intense development based on both rigorous theoretical and H[SHULPHQWDOLQSXW7KHGRPLQDQWIRUFHLQFRVPRORJ\LVJUDYLW\(LQVWHLQ¶V theory of General Relativity (GR), our modern theory of gravity, forms a solid theoretical underpinning of the recent progress. On the experimental side the discovery by Hubble of the expanding universe is the basis of the Big Bang model of a dynamic universe that allows a meaningful discussion of a beginning, and thus an age of the universe. The discovery of the Cosmic Microwave Background (CMB), the relic sea of photons left over from the early universe, further supports the Big Bang model. In the flat Universe where the curvature is zero and the density is the critical density, new hypothetical objects, dark energy and dark matter, are essential to bridge the gap between theory and observation. It is clear that these new ingredients, dark energy and dark matter, arose as a consequence to the false flat universe paradigm. Without them, there would be a fatal contradiction between the observations made by astronomers and the predictions of the big bang theory. The realization from the rotation curves of galaxies that there is considerably more matter with attractive gravity, called dark matter, than we can see via electromagnetic radiation (the visible matter) complicates things. With attractive gravity between the components of the universe the expectation was that the expansion velocity of the universe, the so called Hubble velocity, would decrease with time, i.e. a decelerating universe. The recent discovery that the Hubble velocity is increasing with time, an accelerating universe, was an immense surprise. This led to the postulation ϱ

of the existence of a new component of the universe, dark energy, with some very unusual properties, not the least among which is that it has, in some sense of the word, repulsive gravity that drives the acceleration. We modify the laws of gravity based on a hyperbolic spacetime rather than a flat spacetime [S. A. Mabkhout, Phys. Essays 26,422 (2013)]. Such a modification explains the accelerating expansion of the Universe without need for dark energy and inflation and also explains the flat rotation curve without need for dark matter.

.

Although perspective for nearby objects in hyperbolic space is very nearly identical to Euclidean space (i.e. the Universe locally is approximately flat consistent with local observations), the apparent angular size of distant objects falls off much more rapidly, in fact exponentially. The Universe is globally hyperbolic as we did prove mathematically. Such a solution predicts the equation of state of cosmology,

S S , which seems as a property of

the Hyperbolic Universe. The hyperbolic structure of the space causes the accelerated expansion of the universe. The hyperbolic structure of the space causes the accelerated expansion of the universe equivalent to its negative pressure. The hyperbolic universe inflates exponentially produces an accelerated expansion of the universe without cosmological constant or scalar field. We KDYHVKRZQWKDWJHQHUDOUHODWLYLW\GRHVQ¶WEUHDNGRZQDWODUJHFRVPRORJLFDO scale since it predicts both the accelerated expansion of the universe (without invoking dark energy) and predicts the galaxy flat rotation curve ZLWKRXW LQYRNLQJ GDUN PDWWHU *HQHUDO UHODWLYLW\ GLGQ¶W EUHDN GRZQ DW Planck scale as we had shown [S. A. Mabkhout, Phys. Essays 25, 112 ϲ

(2012)]. In this research we shall prove that the time evolution equation of the universe characteristics the hyperbolic universe and traces its manifold dynamical geometry shouldn`t break down even at the initial Big Bang moment. Our task is to remove the singularity from the mathematical model, represented by the General Relativity Theory and the hyperbolic spacetime, underlying the Big Bang and the Black Hole. Moreover , recession velocities exceed the speed of light in all viable cosmological models for objects with red shifts greater than z ~ 1.5. Locally, the spacetime is flat where the speed of light is the constant "c". The hyperbolic structure of the spacetime could only be recognizable globally. )DVWHU WKDQ OLJKW PRWLRQ RFFXUV RXWVLGH WKH REVHUYHU¶V LQHUWLDO IUDPH ORFDO frame). As the objects are far distant separated as much the spacetime is hyperbolic curved, as faster the velocity of light exceeds the constant speed "c" of light due to flat spacetime. The flat spacetime constraints the domain of applicability of special relativity. In testing the validity of any scientific paradigm, the key criterion is whether measurements agree with what is expected given the paradigm. Our Hyperbolic Universe paradigm agrees with observations and accounts for the acceleration expansion of the universe and the flat rotation curve. In comparison with other models such as modifying gravity models fail to agree with observations or those need additional assumptions like the need for the existence of new gradients such as dark energy or dark matter. Our model seems to be aesthetically pleasing and preferred according to the Criteria for scientific method and Occam's razor as stated below:

ϳ

Criteria for scientific method (i)

The model must fit the data and agree with observations.

(ii)

The model must make predictions that allow it to be tested (falsifiable).

(iii)

The model should be aesthetically pleasing (Occam's razor)

Occam's razor : "If you have two theories that both explain the observed facts, then you should use the simplest until more evidence comes along" "The simplest explanation for some phenomenon is more likely to be accurate than more complicated explanations." "If you have two equally likely solutions to a problem, choose the simplest." "The explanation requiring the fewest assumptions is most likely to be correct."

ϴ

ĐŽŶƚĞŶƚƐ WƌĞĨĂĐĞ

ϱ

Part I: General Relativity Doesn`t Break

ϭϯ

down At Any Scale And Possesses No Singularity 1. Introduction

ϭϱ

2. The dynamical equation of cosmology

ϭϳ

3. The Hyperbolic Geometry of the Universe

Ϯϯ

3.1 Method of solution

Ϯϲ

3.2 Method of solution of the dynamical equations in the presence of the cosmological constant

ϯϱ

3.3 Verification of time evolution equation of

ϯϴ

the universe at Planck scale and current scale 4. General Relativity theory through its

ϰϭ

K\SHUEROLFXQLYHUVHGRHVQ¶WQHHG,QIODWLRQ 4.1. Inflation

ϰϭ

4.2. BICEP2 and Planck space observatory

ϰϲ

4.3. Criticisms (The inflationary paradigm is

ϰϴ

untestable, and scientifically meaningless) 4.4. Hyperbolic expansion of the universe possesses a legitimate inflation ϵ

ϱϭ

5. The Big Bang hyperbolic universe

ϱϰ

possesses nonsingularity 6. Nonsingular Black Holes

ϲϮ

6.1. The Schwarzschild metric

ϲϮ

6.2. Nonsingular Schwarzschild Black Hole

ϲϲ

6.3. Equation of the radial motion in the

ϲϴ

galaxy`s hyperbolic space-time 6.4. The modified Schwarzschild metric in the

ϳϭ

hyperbolic spacetime 6.5. Nonsingular Kerr Rotating Black Hole in

ϳϯ

the hyperbolic spacetime 7. Information loss paradox resolved by nonsingular hyperbolic spacetime

ϳϲ ϳϴ

7.1. Hawking Radiation ϴϬ 7.2. Information Loss Paradox ϴϯ 7.3. Black Hole Complementarity ϴϲ 7.4. Firewall (failure of Complementarity)

ϭϬ

+DZNLQJ(YHQW+RUL]RQ'RHVQ¶W([LVW

ϴϵ ϵϰ

7.6. Laura Mersini-Houghton: Black Holes do not Exist ϵϳ 7.7. Singularity, in Principle, Theoretically doesn't Exist Summary

ϭϬϱ

References

ϭϬϲ

Part II: Modification of the laws of gravity

ϭϭϭ

1. Introduction

ϭϭϮ

2. Method of solution of the dynamical

ϭϭϱ

equations of cosmologyin the presence of the cosmological constant 3. Hyperbolic Universes Does Not Need

ϭϮϮ

Dark Energy 3.1 Incorrect zero pressure-dust universe model

ϭϮϮ

3.2 The accelerating expansion of the

ϭϮϰ

Universe 4. Hyperbolic Universe Does Not Need Dark

ϭϮϴ

Matter 4.1 Flat rotation curve

ϭϭ

ϭϮϴ

5. The Superluminous Speed through the

ϭϰϯ

Hyperbolic 5.1 Doppler redshift and Gravitational

ϭϰϯ

redshift 5.2 The Superluminous Speed

ϭϰϰ

Summary

ϭϰϴ

References

ϭϱϬ

ϭϮ

Part I

General Relativity Doesn`t Breakdown

At Any Scale And Possesses Nonsingularity

ϭϯ

ϭϰ

1. Introduction The Universe is globally hyperbolic as we did prove mathematically [S. A. Mabkhout, Phys. Essays 25, 112 (2012)]. We show that the equation of the hyperbolic time evolution of the universe characteristics the hyperbolic universe and traces its manifold dynamical geometry shouldn`t break down even at the initial Big Bang moment. Hence singularity did not exist at the Big Bang. We developed the Schwarzschild metric in the hyperbolic spacetime. We show it possesses no singularity. The hyperbolic time evolution equation of the universe successfully predicts the Planck length at micro-cosmos scale as well as it predicts the current observed large structure at macro-cosmos scale). The Information Loss Paradox does no longer exist in the nonsingular hyperbolic spacetime. Indeed, in most scientific arenas, singular behavior is viewed as an indication that the theory being used is deficient. It is therefore common to claim that general relativity, in predicting that spacetime is singular, is predicting its own demise, and that classical descriptions of space and time break down at black hole singularities and at the Big Bang. Such a view seems to deny that singularities are real features of the actual world, and to assert that they are instead merely artifices of our current (flawed) physical theories Many physicists and philosophers argue that the singular behavior at the center of black holes and at the beginning of time points to the limit of the domain of applicability of general relativity. We update general relativity theory through our hyperbolic universe (spacetime) solution to the

ϭϱ

dynamical equations of cosmology (Friedmann's equations). We show the hyperbolic spacetime prevents singularity to occur. The hyperbolic universe inflates exponentially produces an accelerated expansion of the universe without cosmological constant or scalar field. We KDYHVKRZQWKDWJHQHUDOUHODWLYLW\GRHVQ¶WEUHDNGRZQDWODUJHFRVPRORJLFDO scale since it predicts both the accelerated expansion of the universe (without invoking dark energy) and predicts the galaxy flat rotation curve ZLWKRXW LQYRNLQJ GDUN PDWWHU *HQHUDO UHODWLYLW\ GLGQ¶W EUHDN GRZQ DW Planck scale as we had shown [S. A. Mabkhout, Phys. Essays 25, 112 (2012)]. In this research we shall prove that the time evolution equation of the universe characteristics the hyperbolic universe and traces its manifold dynamical geometry shouldn`t break down even at the initial Big Bang moment. Our task is to remove the singularity from the mathematical model, represented by the General Relativity Theory and the hyperbolic spacetime, underlying the Big Bang and the Black Hole.

. .

. . .

ϭϲ

2. The dynamical equation of cosmology To derive the dynamical equation of cosmology, we should combine Einstein field equations 5NO

JNO } Q*7NO

with the isotropic homogeneous Robertson- Walker's line-element:

GU ¬ U G R U VLQ R G G GV GW 5 W ® NU

where [

W [ U [ R [ G

The corresponding components of the metric tensor are:

J J J

5 J NU

J 5 U J

J VLQ R J J Now according to the affine connection: M

(NO

sJ sJ ¬ sJ JMT BON TNO NOT s[ s[ ® s[

ϭϳ

We compute:

5 55 ( 5 NU ( 55U VLQ R ( 55U NU ( ( U NU ( VLQ R( NU ( ( ( VLQ R ( FRW R U (

$OORWKHUFRPSRQHQWVRIīHLWKHUYDQLVKRUIROORZIURPWKHV\PPHWU\ M

(ON

(MNO

A dot denotes differentiation with respect to time .Latin indices run over the values 1, 2 and 3. Note the Ricci tensor: 5NO

s(MNO s[

M

s(MNM s[

O

(TNO (MTM (TNM(MTO

we calculate the nonzero components of the Ricci tensor, the non-vanishing components of which are easily found to be:

55 5 N E ML 5 5 5 5 5LM

ϭϴ

55 5 N NU 5 U 55 5 N 5 U 55 5 N VLQ R 5

and the Ricci scalar is then

} J NO 5NO J 5 J 5 J 5 J 5

5 N ¬ 5 ¬ } 5 5 ® 5 ®

Turning our attention now to the cosmological fluid, we assume that it is described by an ideal fluid. The fluid will be at rest in co-moving coordinates. The four-velocity is then 8

N

According to the energy momentum tensor

7 SJNO S S 8N8O

NO

JNO8 N8 O

:KHUHSLVWKHSUHVVXUHDQGȡLVWKHHQHUJ\GHQVLW\RIWKHFRVPRORJLFDOIOXLG The energy-momentum tensor ϭϵ

O

7N

SENO S S 8 O8N

8 O8 O

becomes

7 NO

S

J S

J S

¬ J S®

7 GLDJ S S S S

N O

7 7NN S S

Consider the zero component of the conservation of energy equation:

N7N N s N7N ( NM 7M (MN 7MN 5 s S S S

5 5 S S S 5

Rewrite Einstein field equations 5NO

JNO } Q*7NO

The NO equation is ϮϬ

5NO JNO } Q*7NO 5 J} Q*7 5 5 5 ¬ N ¬ q Q*S 5 5 5 ® 5 ® Q 5 N *S5 There is only one distinct equation from NO LM , due to isotropy

5NO JNO } Q*7NO 5 J} Q*7 5 N 5 U q U 55 5 5 ¬ N ¬ 5 5 5 Q*S5 U ® ® 5 U U N Q*S5 U = 55U 5 N Q*S5 = 55 Ϯϭ

We get the following two dynamical equations of cosmology (Friedmann`s equation):

5 N Q*S 5 5 N Q S 55

ϮϮ

.

3. The Hyperbolic Geometry of the Universe The shape of the universe depends on the curvature k (the field strength) according to the following three cases (1) If K=0 then the universe is flat and open. (2) If K>0 then the universe is spherical and closed. (3) If K/S F X /S F @GX /S F X GW = ¨ >/S F X /S F @GX /S F X ¨ GW = ¨ /S F X GX /S F ¨ GX /S F X ¨ GW

= F VLQK X /S F X /S F X /S F VLQK X F F W = W X /S F X /S F VLQK X /S F F = W 5 /S F 5 /S F VLQK 5 /S F F

dŽĨŝŶĚĐϮǁĞĂƐƐƵŵĞƚсƚƉǁŚĞŶZс>Ɖ͘

ϯϭ

W S /S /S F /S /S F VLQK /S /S F F

W S /S F /S F VLQK F F W S F/S /S F VLQK F F

W S / S F = /S F F/S /S F VLQK F F To find F

F /S F F /S F VLQK F OLPF VLQK F F l

VLQK F OLP F l F [ VLQK [ OLP [ld [ld [ [ OLP [ld [ [ OLP

F /S F F SXW F = F

ϯϮ

W 5 5/S F

SXW F = 5W

Clearly, the solution satisfies the differential equation (2) in its pressure less form

55 5 N 5 W 5 5 N

W

Finally, we reach to the solution of the second dynamical equation of cosmology, the space-space component (2),in it is pressure less form, to be

5 W ZKLFKUHIOHFWVWKHVDPHVROXWLRQIRUHTXDWLRQ IRUȡ VLQFH

5 N Q S5 SXW S N = 5 5W Note WKDWWKLVVROXWLRQLVFRQVLVWHQWZLWKRXUVROXWLRQ IRUȡ VLQFH

ϯϯ

5 QS VLQK W QS 5 OLP QS VLQK W QS S l

HW W 5 OLP Sl

HW QS ¬ QS W ®

QS

W 5

5W

3.2 Method of solution of the dynamical equations in the presence of the cosmological constant: (i) The empty spacetime case: The dynamical equations of cosmology for the empty space-time in the presence of cosmological constant are given by:

5NO JNO 5 Q7NO 5 N -5 5 N -5 55 - QS

YDFXXP

5 N QS YDFXXP 5 5 N QS YDFXXP 5 55

The vacuum energy is constant since it is not dilute with time, hence ϯϰ

5 N QS YDFXXP 5 5

¨ GU

W

U N QS

YDFXXP

QS

YDFXXP

¨ GU

5 L N QS YDFXXP VLQK W QS YDFXXP = N

5 QS YDFXXP VLQK W QS YDFXXP Substitute the solution of equation (5) into equation (6) in order to be consistent, we get

5 QS YDFXXP VLQK W QS YDFXXP 5 FRVK W QS YDFXXP 5 QS YDFXXP VLQK W QS YDFXXP QS YDFXXP 5

ϯϱ

55 5 N -5 55 5 QS YDFXXP 5 5 QS YDFXXP FRVK W QS YDFXXP QS YDFXXP 5 5 QS YDFXXP VLQK W QS YDFXXP QS YDFXXP 5 5

QS YDFXXP QS YDFXXP 5 QS YDFXXP5

QS YDFXXP 5 QS YDFXXP 5 (ii) The nonempty spacetime case: The dynamical equations of cosmology for the nonempty space-time in the presence of cosmological constant are given by :

5 N -5 Q S5 55 5 N -5 Q S5 - QS

YDFXXP

YDFXXP 5 N QS 5 Q S5 YDFXXP 55 5 N QS 5 Q S5

Since the cosmological constant represents about 74% of the matter of the universe, it dominates

S

YDFXXP

S

. So it is plausible that S

approximately constant, hence ϯϲ

YDFXXP

S

is

YDFXXP 5 N QS 5 Q S 5

YDFXXP 5 N Q S S 5

5 Q S

YDFXXP

S VLQK W Q S

YDFXXP

S

Substitute the above solution of equation (6) into (7),we get YDFXXP 5 N QS 5 Q S5 YDFXXP 55 5 N QS 5 Q S5

Q S YDFXXP S 5 QS S YDFXXP S S

YDFXXP

S

YDFXXP

5 Q S5 .

= S S So the cosmological constant (the vacuum energy) disappeared in the solution of the second differential eq (8). Just the ordinary energy density state S M remains in the Hyperbolic Universe to derive the accelerating expansion equivalent to its negative pressure. Hyperbolic Universe involves zero9 cosmological constant (the vacuum energy). The negative pressure

S S M is the property of the hyperbolic structure of the Universe. Moreover the hyperbolic evolution of the Universe predicts the large structure of the observable Universe 10 28cm associated with 14 × 109 yr as shown in Eq. (9) below.

ϯϳ

3.3 Verification of the time evolution equation of the universe at Planck scale and current scale Planck scale: Planck time is the time needed by something moving at the speed of light to travel Planck length / S

.

Regarding, quantum theory we have the following Planck length= /S *K F q FP Planck time = WS /S F *K F q V Planck mass = 1.2×1025eV/c2 Planck density = M/V = M/L3 = 1.2×1025(eV/c2)/(1.6×10-33cm)3 There is no any connection between Quantum theory which deals with particles

that have approximately

negligible

gravity, and general

relativity theory, which deals with massive objects that have enormous gravity. Each theory stands on it is own concepts and grounds with no chance of meeting at any point. Physicists have been trying to accomplish the unification of the microcosm and the macrocosm.

.

Surprisingly , when I tried to verify my solution at the Planck`s scale of the universe (I used the Planck density and Planck time as given data and I assumed that the Planck length is unknown), the result was exactly Planck length. My solution

.

ϯϴ

5 M QS M VLQK W M QS M

has a special feature, since it is derived from General relativity theory and exactly predicts Planck Length from Quantum theory . Let us apply equation (3) to the data of Planck scale of the universe. To do this, we substitute a given Planck time and Planck density in equation (3) while we assume Planck length is unknown. Note that in geometrical units The speed of light c = 1 The gravity G=1 1 sec = 2.997×1010cm 1 gram = 7.425×10-29cm 1 eV = 1.324×10-61cm. Planck density = M/V = M/L3 = 1.2×1025(eV/c2)/(1.6×10-33)3 =1.2×1025(1.324×10-61cm)/(1.6×10-33)3=3.8789×1062cm Substitute the above data in equation (3) 5S

QSS VLQK WS QSS

5S Q qq VLQKq q q Q qq = 0.175423 × 10-31 × sinh0.092255888 ϯϵ

= 0.175423×10-31 × 0.092386811 = 1.62 × 10-33 cm = Lp = Planck length. Hence 5S

QSS VLQK W S QSS /S

*K

F

The calculations above prove the validity of equation (3) at Planck scale since it preserves Planck Length, which verifies my solution and confirms that the curvature k must be negative. Moreover the previous method provide a first connection relates General Relativity Theory to Quantum Theory. Since the time evolution equation of the universe is derived from the General relativity theory, exactly predicts Planck Length from Quantum theory. Hence, General Relativity theory doesn`t collapse at micro-scale. current scale: The Hyperbolic Universe inflates, through the hyperbolic time evolution equation legitimately to 1028 cm, very consistent with the

current observable universe. 3

Note that in geometrical units 1 sec = 2.997×1010cm 1 gram = 7.425×10-29cm 1 yr = 3.16×107 s ϰϬ

The energy density now SQRZ =10-31 g/cm3 =7.425×10-60cm-2 The age of the Universe (approximately) WQRZ =14×109 yr= q cm Substitute the above data in the hyperbolic time evolution equation of the Universe, yields

5 M QS M VLQK ¡W M QS M ¯° ¢ ± 5QRZ QSQRZ VLQK ¡WQRZ QSQRZ ¯° ¢ ±

5QRZ Q qq q VLQK ¡ q q Q qq ¯° ¢ ± 5QRZ q q VLQK 5QRZ q FP

The Hyperbolic Universe grows exponentially preserve a legitimate inflation covers the current large structure.

ϰϭ

4. General Relativity theory through its K\SHUEROLFXQLYHUVHGRHVQ¶WQHHG,QIODWLRQ 4.1. Inflation Inflation arose as a consequence to the false flat universe paradigm. Consider a photon moving along a radial trajectory in a flat universe. A radial null path obeys

R GV GW 5 W GU W

U

GW

¨ 5 W

WH

For matter dominated component of energy : 5 r W

Hubble constant now (at W ) is + + W

5 W

5 W

So the age of the universe now is

W

+ NP V 0SF ¶¶¶¶¶¶ l *\U +

which is inconsistent compared to the age of the oldest stars whose age is about 12 Gyr in our galaxy. Equations due to the flat universe doesn`t fit the data. The flat universe must be updated by inflation, say. Instead of

5 r W , the scale factor should growths exponentially 5 r H +W ϰϮ

In physical

cosmology,

inflation is

the

theorized

extremely

rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary

epoch comprises

the

first

part

of

the electroweak

epoch following the grand unification epoch. It lasted from 10í36 seconds after the Big Bang to sometime between 10í33 and 10í32 seconds Following the inflationary period, the universe continued to expand, but at a slower rate15. Inflation proposed to fabricate answers to the classic conundrum of the Big Bang cosmology: The horizon problem: is the problem of determining why the universe appears statistically homogeneous and isotropic in accordance with the cosmological principle. The cosmic microwave background is the cooled remains of the radiation density from the radiation-dominated phase of the Big Bang. Observations of the cosmic microwave background show that it is amazingly smooth in all directions, in other words, it is highly isotropic thermal radiation. The temperature of this thermal radiation is 2.73° Kelvin. The variations observed in this temperature across the night sky are very tiny. Radiation can only be so uniform if the photons have been mixed around a lot, or thermalized, through particle collisions. However, this presents a problem for the Big Bang model. Particle collisions cannot move information faster than the speed of light. But in the expanding Universe that we appear to live in, photons moving at the speed of light cannot get from one side of the Universe to the other in time to account for this observed isotropy in the thermal radiation. The horizon size represents the distance a photon can travel as the Universe expands. The horizon size of our Universe ϰϯ

today is too small for the isotropy in the cosmic microwave background to have evolved naturally by thermalization. So that's the horizon problem. The flatness problem : that the density of matter in the universe was comparable to the critical density necessary for a flat universe. The Universe as observed today seems to enough energy density in the form of matter and cosmological constant to provide critical density and hence zero spatial curvature. The Einstein equation predicts that any deviation from flatness in an expanding Universe filled with matter or radiation only gets bigger as the Universe expands. So any tiny deviation from flatness at a much earlier time would have grown very large by now. If the deviation from flatness is very small now, it must have been immeasurably small at the start of the part of Big Bang we understand. So why did the Big Bang start off with the deviations from flat spatial geometry being immeasurably small? This is called the flatness problem of Big Bang cosmology. Whatever physics preceded the Big Bang left the Universe in this state. So the physics description of whatever happened before the Big Bang has to address the flatness problem. Inflationary models also solve the horizon problem. The vacuum pressure accelerates the expansion of space in time so that a photon can traverse much more of space than it could in a spacetime filled with matter. To put it another way, the attractive force of matter on light in some sense slows the light down by slowing down the expansion of space itself. In an inflationary phase, the expansion of space is accelerated by vacuum pressure from the cosmological constant, and light gets farther faster because space is expanding faster. If there were an inflationary phase of our Universe before ϰϰ

the radiation-dominated era of the Big Bang, then by the end of the inflationary period, light could have crossed the whole Universe. And so the isotropy of the radiation from the Big Bang would no longer be inconsistent with the finiteness of the speed of light.

.

Inflationary universe: matter and radiation are gravitationally attractive, so in a maximally symmetric spacetime filled with matter, the gravitational force will inevitably cause any lumpiness in the matter to grow and condense. That's how hydrogen gas turned into galaxies and stars. But vacuum energy comes with a high vacuum pressure, and that high vacuum pressure resists gravitational collapse as a kind of repulsive gravitational force. The pressure of the vacuum energy flattens out the lumpiness, and makes space get flatter, not lumpier, as it expands. So one possible solution to the flatness problem would be if our Universe went through a phase where the only energy density present was a uniform vacuum energy. If this phase occurred before the radiation-dominated era, then the Universe could evolve to be extraordinarily flat when the radiation-dominated era began, so extraordinarily flat that the lumpy evolution of the radiation- and matterdominated periods would be consistent with the high degree of remaining flatness that is observed today. This type of solution to the flatness problem was proposed in the 1980s by cosmologist Alan Guth. The model is called the Inflationary Universe. In the Inflation model, our Universe starts out as a rapidly expanding bubble of pure vacuum energy, with no matter or radiation. After a period of rapid expansion, or inflation, and rapid cooling, the potential energy in the vacuum is converted through particle physics processes into the kinetic energy of matter and radiation. The Universe heats up again and we get the standard Big Bang. So an inflationary phase before ϰϱ

the Big Bang could explain how the Big Bang started with such extraordinary spatial flatness that it is still so close to being flat today. How does Inflation Work? Not like vacuum energy today, at the very early universe elementary particle interactions themselves could generate an inflationary expansion. The vacuum energy that drives the rapid expansion in an inflationary cosmology comes from a scalar field that is part of the spontaneous symmetry breaking dynamics of some unified theory particle theory, say, a Grand Unified Theory or string theory. This scalar field is sometimes called the inflaton. The equation of motion for this field in the de Sitter metric above is and the Einstein equation with a scalar field density becomes

The conditions for inflationary behavior require that the scalar field time derivatives are small compared to the potential, so that most of the energy of the scalar field is in potential energy and not kinetic energy4

ϰϲ

4.2. BICEP2 and Planck space observatory According to the team at the BICEP2 South Pole telescope, the detection is at the 5±7 sigma level, so there is less than one chance in two million of it being a random occurrence. The results were hailed as proof of the Big Bang inflationary theory and its progeny, the multiverse. The BICEP2 team identified a twisty (B-mode) pattern in its maps of polarization of the cosmic microwave background, concluding that this was a detection of primordial gravitational waves. Now, serious flaws in the analysis have been revealed that transform the sure detection into no detection. The search for gravitational waves must begin anew. The problem is that other effects, including light scattering from dust and the synchrotron radiation generated by electrons moving around galactic magnetic fields within our own Galaxy, can also produce these twists. Two groups of scientists announced that a tantalizing signal "smoking gun" evidence of dramatic cosmic expansion just after the birth of the universe was actually caused by something much more mundane: interstellar dust. In the cosmic inflation announcement, which was unveiled in March 2014, scientists with the BICEP2 experiment, claimed to have found patterns in light left over from the Big Bang that indicated that space had rapidly inflated at the beginning of the universe, about 13.8 billion years ago. The discovery also supposedly confirmed the existence of gravitational waves, theoretical ripples in spacetime. Scientists with the European Space Agency said that data from the agency's Planck space observatory has revealed that interstellar dust caused more than half of the signal detected.

ϰϳ

BICEP2, Planck and Keck all study the cosmic microwave background (CMB), or light that is left over from the Big Bang, and which can be seen in every direction in the sky. One feature of the CMB that these experiments study is its polarization, or the orientation of the light waves. If inflation did occur when the universe was born, it would have perturbed the fabric of the universe ²spacetime ² creating what are known as gravitational waves. These waves would have then created swirls in the polarization of the CMB, or what are called B-modes. Thus, the discovery of these B-modes would have meant both confirmation of inflation and evidence of gravitational waves. But the Planck results show that the light from dust is significant over the entire sky ² including the region where BICEP2 purportedly observed B-modes. While BICEP2 only sees the sky in one wavelength of light, Planck observes the universe in nine wavelength channels, which help this instrument separate the CMB signal from the background. When the dust is accounted for, the signal identified by BICEP2 becomes too faint to be considered significant. 4.3. Criticisms (The inflationary paradigm is fundamentally untestable, and hence scientifically meaningless) There are hundreds of models of inflation, each with its own prediction about how fast the universe expanded and no one is much likely than the other. The BICEP2 incident has also revealed a truth about inflationary theory. The common view is that it is a highly predictive theory. If that was the case DQG WKH GHWHFWLRQ RI JUDYLWDWLRQDO ZDYHV ZDV WKH µVPRNLQJ JXQ¶ SURRI RI ϰϴ

inflation, one would think that non-detection means that the theory fails. Such is the nature of normal science. Yet some proponents of inflation who celebrated the BICEP2 announcement already insist that the theory is equally valid whether or not gravitational waves are detected. How is this possible? The answer given by proponents is alarming: the inflationary paradigm is so flexible that it is immune to experimental and observational tests. First, inflation is driven by a hypothetical scalar field, the inflaton, which has properties that can be adjusted to produce effectively any outcome. Second, inflation does not end with a universe with uniform properties, but almost inevitably leads to a multiverse with an infinite number of bubbles, in which the cosmic and physical properties vary from bubble to bubble. The part of the multiverse that we observe corresponds to a piece of just one such bubble. Scanning over all possible bubbles in the multiverse, everything that can physically happen does happen an infinite number of times. No experiment can rule out a theory that allows for all possible outcomes. Hence, the paradigm of inflation is unfalsifiable and scientifically meaningless. This may seem confusing given the hundreds of theoretical papers on the predictions of this or that inflationary model. What these papers typically fail to acknowledge is that they ignore the multiverse and that, even with this unjustified choice, there exists a spectrum of other models which produce all manner of diverse cosmological outcomes. Taking this into account, it is clear that the inflationary paradigm is fundamentally untestable, and hence scientifically meaningless5

.

David Parkinson at the University of Queensland in Australia and his colleagues decided to look at the nature of those apparent gravitational ϰϵ

waves to see if they were the type of waves predicted by inflation. And they weren't. Counter to what the BICEP2 collaboration said initially, Parkinson's analysis suggests the BICEP2 results actually rule out any reasonable form of inflationary theory. Most inflationary models require that as you look at larger and larger scales of the universe, you should see stronger and stronger gravitational waves. Cosmologists call that a "gravitational wave spectrum". "What inflation predicted was actually the reverse of what we found," says Parkinson. How many inflationary models does it rule out? "Most of them, to be honest." Nobel laureate Brian Schmidt at the Australian National University in Canberra, who has been critical of the theory of inflation, says he expects that further analysis will confirm that no gravitational waves were observed at all. "But on the other hand, if BICEP2 is shown to be correct, it's exciting," says Schmidt. "And it does potentially break standard inflation and therefore you are testing inflation and showing its wrong." Paul Steinhardt of Princeton University, who helped develop inflationary theory but is now scathing of it, says this is potentially a blow for the theory, but that it pales in significance with inflation's other problems. Steinhardt says the idea that inflationary theory produces any observable predictions at all ± even those potentially tested by BICEP2 ± is based on a simplification of the theory that simply does not hold true. "The deeper problem is that once inflation starts, it doesn't end the way these simplistic calculations suggest," he says. "Instead, due to quantum physics it leads to a multiverse where the universe breaks up into an infinite number of patches. The patches explore all conceivable properties as you go from patch to patch. So that ϱϬ

means it doesn't make any sense to say what inflation predicts, except to say it predicts everything. Steinhardt says the point of inflation was to explain a remarkably simple universe. "So the last thing in the world you should be doing is introducing a multiverse of possibilities to explain such a simple thing," he says. "I think it's telling us in the clearest possible terms that we should be able to understand this and when we understand it it's going to come in a model that is extremely simple and compelling. And we thought inflation was it ± but it isn't."6 4.4. Hyperbolic expansion of the universe possesses a legitimate inflation Inflation fabricates an initial conditions to predict a flat Universe. The flat universe assumption arose due to the local observations which show that the universe is approximately flat restricted to the existence of pointless dark energy and dark matter and illegitimately generalized to the whole global geometry of the universe. Although general relativity says space-time locally is approximately flat but that doesn't imply it must be globally flat. Such problematic flat universe paradigm needs inflation to stay alive. The hyperbolic evolution of the Universe exhibits rational and reasonable inflation covers the large structure, isotropic and homogeneity in accordance with the cosmological principle. So, the flatness problem does no longer exist in the Hyperbolic Universe. Rather than the unjustified magical inflation that occurs within sometime between 10í33 and 10í32 seconds cause the universe expand to an order 10 50 , we did prove the hyperbolic geometry of the universe. Such a Hyperbolic Universe possesses a reasonable hyperbolic inflation through its whole age. ϱϭ

The Hyperbolic Universe inflate, through the hyperbolic time evolution equation (5), legitimately to 1028 cm, very consistent with the current observable universe7 Note that in geometrical units 1 sec = 2.997×1010cm 1 gram = 7.425×10-29cm 1 yr = 3.16×107 s The energy density now SQRZ =10-31 g/cm3 =7.425×10-60cm-2 The age of the Universe (approximately) WQRZ =14×109 yr= q cm Substitute the above data in the hyperbolic time evolution equation of the Universe, yields

5 M QS M VLQK ¡W M QS M ¯° ¢ ± 5QRZ QSQRZ VLQK ¡WQRZ QSQRZ ¯° ¢ ± 5QRZ Q q q q VLQK ¡ q q Q q q ¯° ¢ ± 5QRZ q q VLQK 5QRZ q FP

The Hyperbolic Universe grows exponentially preserve a legitimate inflation covers the current large structure. Hence, the horizon problem also ϱϮ

does no longer exist, since the backward exponential contraction re-put both sides of the Universe at causal contact. Penrose said if N v R , then inflation is out ! 30 Instead of unphysical inflation epoch the Hyperbolic Universe grows exponentially preserve a legitimate inflation covers the current observed large structure (1028 cm).

ϱϯ

5. The Big Bang hyperbolic universe possesses nonsingularity The Universe is globally hyperbolic as we did prove mathematically [S. A. Mabkhout, Phys. Essays 25, 112 (2012)]. We show that the equation of the time evolution of the universe 5W QS M VLQK W QS M characteristics the hyperbolic universe and traces its manifold dynamical geometry shouldn`t break down even at the initial Big Bang moment. An initial-value problem: Given the state of a system at some moment in time, what will be the state at some later time? Future events can be understood as consequences of initial conditions plus the laws of physics. Could the dynamical nature of the spacetime background break down an initial-value formulation in general relativity? In general relativity, a singularity is a place that objects or light rays can reach in a finite time where the curvature becomes infinite, or spacetime stops being a manifold. Singularities can be found in all cosmological solutions which don't have scalar field energy or a cosmological constant. Curvature is associated with gravity and hence curvature singularities correspond to "infinitely strong gravity." There are several possibilities of how such infinitely strong gravity can manifest itself. For instance, it could be that the energy density becomes infinitely large - this is called a "Ricci singularity", As an example of a Ricci singularity, the evolution of energy density in a universe described by a big bang model. As you go towards the left - corresponding to earlier and earlier instances of cosmic time zero - the ϱϰ

density grows beyond all bounds and at cosmic time zero - at the big bang it was infinitely high. A path in spacetime is a continuous chain of events through space and time. While there are competing definitions of spacetime singularities, the most central, and widely accepted, criterion rests on the possibility that some spacetimes contain incomplete paths. Indeed, the rival definitions (in terms of missing points or curvature pathology) still make use of the notion of path incompleteness. While path incompleteness seems to capture an important aspect of the intuitive picture of singular structure, it completely ignores another seemingly integral aspect of it: curvature pathology. If there are incomplete paths in a spacetime, it seems that there should be a reason that the path cannot go farther. The most obvious candidate explanation of this sort is something going wrong with the dynamical structure of the spacetime, which is to say, with the curvature of the spacetime. This suggestion is bolstered by the fact that local measures of curvature do in fact blow up as one approaches the singularity of a standard black hole or the big bang singularity. However, there is one problem with this line of thought: no species of curvature pathology we know how to define is either necessary or sufficient for the existence of incomplete paths. "At the heart of all of our conceptions of a spacetime singularity is the notion of some sort of failing: a path that disappears, points that are torn out, spacetime curvature that becomes pathological. However, perhaps the failing lies not in the spacetime of the actual world, but rather in the theoretical description of the spacetime. That is, perhaps we shouldn't think that general relativity is accurately describing the world when it posits singular structure! Indeed, in ϱϱ

most scientific arenas, singular behavior is viewed as an indication that the theory being used is deficient. It is therefore common to claim that general relativity, in predicting that spacetime is singular, is predicting its own demise, and that classical descriptions of space and time break down at black hole singularities and at the Big Bang. Such a view seems to deny that singularities are real features of the actual world, and to assert that they are instead merely artifices of our current (flawed) physical theories" 3 Many physicists and philosophers resist that singularities are real. Some argue that singularities are too repugnant to be real. Others argue that the singular behavior at the center of black holes and at the beginning of time points to the limit of the domain of applicability of general relativity. Note that the hyperbolic universe inflates exponentially produces an accelerated expansion of the universe without cosmological constant or scalar field. We have shown [S. A. Mabkhout, Phys. Essays 26,422 (2013)] that general relativity doesn't break down at large cosmological scale since it predicts both the accelerated expansion of the universe (without invoking dark energy) and predicts the galaxy flat rotation curve (without invoking dark matter). General relativity didn't break down at Planck scale as we had shown [S. A. Mabkhout, Phys. Essays 25, 112 (2012)]. In this research we shall prove that the time evolution equation of the universe characteristics the hyperbolic universe and traces its manifold dynamical geometry shouldn`t break down even at the initial Big Bang moment. Our task is to remove the singularity from the mathematical model, represented by the General Relativity Theory and the hyperbolic spacetime, underlying the Big

ϱϲ

Bang and the Black Hole. Our main point is to examine whether the state point

5%% S%% W%% d

Constitutes a singular point in the manifold? Is it really a missing point of the manifold? Does the local measure of curvature blow up as one approach this point? Does the density grow beyond all bounds, infinitely high as one approach this point? Our hyperbolic universe is a manifold weaved by the time evolution equation of the universe since the Big Bang 5M

QS M VLQK ¡W M QS M ¯° ¢ ±

Which reflects the structure of the manifold whether it possesses a singular point or not ? If we assume the density S M and the time W M runs independently from each other, we may evaluate the limit at the Big Bang

S W S M

M

%%

W%% l d

5%% QS%% VLQK ¡W%% QS%% ¯° ¢ ± 5%% OLP QS%% VLQK ¡W%% QS%% ¯° ¢ ± S% % W% % l d

5%%

OLP

S% % W% % l d

W%%

VLQK ¡W%% QS%% ¯° ¢ ± W%% QS%%

VLQK ¡W%% QS%% ¯° ¯° ¡ ¢ ± 5%% OLP ¡ OLP W%%

° S% % l d ¡ W% % l

W%% QS%% ° ¢ ± ϱϳ

5%% OLP < q > S% % l d

VLQK ¡W%% QS%% ¯° ¯° ¡ ¢ ± ¡ OLP W%%

° 5%% W%OLP % l ¡ S% % l d

W%% QS%% ° ¢ ± 5%% OLP d W% % l

The limit does not exist since it is not unique. Let us treat the limit from a different point of view, namely the dependent evolution for both the density

S M and the time W M . It is well known that the time evolution equation of the universe successfully predicts the Planck length at micro-cosmos scale as well as it predicts the current observed large structure at macro-cosmos scale. 1 sec = 2.997×1010cm Planck length= /S *K F q FP Planck time = WS /S F *K F q V Planck density= =3.8789×1062cm Substitute the above data in the time evolution equation of the universe at Planck scale

5S QSS VLQK QSS WS

5S Q qq VLQK Q qq qq q q = ϱϴ

0.175423 × 10-31 × sinh0.092255888 = 0.175423×10-31 × 0.092386811 = 1.62 × 10-33 cm = Lp = Planck length. Hence

5S QSS VLQK QSS W S /S

*K

F

The energy density now SQRZ =10-31 g/cm3 =7.425×10-60cm-2 The age of the Universe (approximately) WQRZ

ð \U q FP

Substitute the above data in the hyperbolic time evolution equation of the Universe, yields

5 M QS M VLQK ¡W M QS M ¯° ¢ ± 5QRZ QSQRZ VLQK ¡WQRZ QSQRZ ¯° ¢ ± 5QRZ Q qq q VLQK ¡ q q Q q q ¯° ¢ ± 5QRZ q q VLQK 5QRZ q FP

ϱϵ

Now we are interesting to explore how both the density S M and the time

WM

are dependently evolved? Consider the factor W M S M appears in the time

evolution equation of the Universe. Calculate the value of W M S M at the given two well known sets of data, namely the Planck scale and the current scale:

WS SS q q q q WQRZ SQRZ q q q The two values are approximately equal no matter how large the difference between the two states, which is of order . Hence it is very reasonable that W M S M remains approximately constant through the whole evolution of the cosmos, even at the Big Bang. The infinitely large density is struggled by the infinitesimally small time and vice versa, in our mathematical model. This process prevents the scale factor from blows up by the infinitely large density. Since the data at Planck scale is accurate, we assume W M S M . Hence

5%%

¡¢ W% %

.

W%% ¯ OLP ± S% % W% % ¯° l ¢ ±

VLQK ¡ Q ¯° ¯° ¡ ¢ ± 5%% OLP ¡ W%%

° W % % l Q ° ¡ ¢ ± ¯ 5%% OLP ¢ W%% q ± W % % l

ϲϬ

VLQK ¡W%% S%% Q ¯° ¢ ± W%% S%% Q

The limit exists. The manifold consists of its limiting point and hence it is complete. Thus there exists a continuous path governs the time evolution of the Universe since the Big Bang. Hence the Big Bang is nonsingular.

ϲϭ

6. Nonsingular Black Holes 6.1. The Schwarzschild metric In General Relativity, the unique spherically symmetric vacuum solution is the Schwarzschild metric. In spherical coordinates

W U R G , the metric is

given by

*0 ¬ *0 ¬ GV GW GU U G8 U ® U ® ZKHUH G8 GR VLQ RGG We are interested in the solution outside a spherical body, Einstein`s equation in vacuum

5NO

The Minkowski metric ( flat spacetime metric) in polar coordinates GV

GW GU U G8

The Schwarzschild metric in the spherically symmetric Put

GV HB U GW H C U GU HM U U G8

U HM U U

,then ϲϮ

GM ¬ GU HM U GU HM U UGM U U HM U GU GU ®

GV H

B U

GM ¬ GW U H C U M U GU U G8 GU ®

Relabeling

U lU

GM ¬ C U M U

l H C U

U GU ® H

We get

GV HB U GW H C U GU U G8

Let us now use Einstein`s equation to solve for the functions

B U DQG C U

The Christoffel symboles are given by

sJ sJ ¬ sJ (MNO JMT BON TNO NOT s[ s[ ® s[

JWW H B JUU H C JRR U JGG U VLQ R JWW H B J UU H C JRR U JGG U FRV HF R

ϲϯ

(WWU s U B

(UUU s U C

U C U R G (GG UH VLQ R (GG VLQ R FRV R (RG FRW R (RUR

U

(WWU H BC s U B (RRU UH C

(GUG

From these we get the following components of the Riemann tensor

W 5UWU s U Bs U C s U s U B

5RWWR UH C s U B 5GW WG UH C VLQ Rs U B 5RUUR UH C s U C

5GU UG UH C VLQ Rs U C R H C VLQ R 5GRG

Taking the contraction yields the Ricci tensor: M 5NO 5NMO

5WW 5WMMW 5WUWU 5WRRW 5WGGW

J UU JWW 5UWUW J RR JWW 5RWWR JGG JWW 5GW WG ¯ H BC ¡s UB s U B s U Bs U C s U B ° ¡¢ °± U

ϲϰ

5UU s UB s U B s U Bs U C s U C U C 5 H ¢U s U C s U B ¯± RR 5GG VLQ R 5RR We would like to set the Ricci tensor equal to zero

H C B 5WW 5UU s U B s U C

U = B C F We can set this constant equal to zero by rescaling our time coorinate

W l HF W = B C 5RR = H B Us U B This is equivalent to

s U UH B

5V U LV FRQV WDQ W

HB 5V

ϲϱ

5V ¬ 5V ¬ GW GU U G8 GV U ® U ®

5V is called Schwarzschild radius . In the weak ±field limit its found that

*0 ¬ JWW U ® = 5V *0

*0 ¬ *0 ¬ GV GW GU U G8 ® ® U U

6.2. Nonsingular Schwarzschild Black Hole The black hole is an object whose escape velocity exceeds the speed of light c. The idea goes back to natural scientist and Anglican rector John Michell, who in 1783 observed that an object with the density of the Sun but 500 times its radius would be a black hole. The nonrelativistic Newtonian equations he used give the correct relativistic formula for the black hole radius. His argument can be summarized beginning with the formula for the energy E of a particle of mass m moving in the gravitational potential of a spherical mass M

*P0 ( PY U where v is velocity, r is radial distance, and G is Newton constant. A particle just barely escapes to infinity if E = 0; if it has an initial velocity v = c, the condition of marginal escape determines an initial radius R(M) = 2GM/c2, now called the Schwarzschild radius for mass M. For an Earth- sized mass, R ϲϲ

is approximately 1 cm; for a solar mass, it is about 3 km31. Our current classical understanding of gravity is via general relativLW\ DQG (LQVWHLQ¶V equations. I1916, mere weeks after their final formulation, Karl Schwarzschild presented a solution giving the gravitational field of a spherically symmetric mass. That Schwarzschild solution is the simplest possible black hole; figure 1 illustrates its basic features. ,Q (LQVWHLQ¶V theory, gravity is described through the curvature of spacetime, and in the center of the black hole, the curvature goes to infinity. That infamous singularity indicates a breakdown of physics, but one far removed from scrutiny. In particular, since in classical physics nothing escapes the region within the so-called event horizon located at R, the singularity has no effect outside the black hole. The need for physics to smooth out the singularity is nonetheless one of the motivators for pursuing a quantum theory of gravity. Many feel that a correct quantum description will resolve such singularities. One of the most remarkable features of relativistic black holes is that they are purely gravitational entities. A pure black hole spacetime contains no PDWWHUZKDWVRHYHU,WLVD³YDFXXP´VROXWLRQWRWKH(LQVWHLQILHOGHTXDWLRQV which just means that it is a solution of Einstein's gravitational field equations in which the matter density is everywhere zero. (Of course, one can also consider a black hole with matter present.) In pre-relativistic physics we think of gravity as a force produced by the mass contained in some matter. In the context of general relativity, however, we do away with gravitational force, and instead postulate a curved spacetime geometry that produces all the effects we standardly attribute to gravity. Thus a black hole LVQRWD³WKLQJ´ in spacetime; it is instead a feature of spacetime itself.

ϲϳ

6.3. Equation of the radial motion in the galaxy`s hyperbolic space-time We develop an equation describes the speed up motion in the hyperbolic space-time and predicts the flat rotation curve. To do this, I will follow the following strategy 1- Seek for an equation Y I U such that Y OLP I U U l

U

¬ D

l 2- Y I U ¶¶¶l N ¶¶¶ ® O DUJ H U

U ld

N D

3- I guess the required equation , that fits the data, should be

Y I U H

U

¬ N , U D®

4- The final step in the mathematical problem solving method is to prove the conjecture

Y I U H

U

¬ N U D®

To find such an equation of the radial motion in the galaxy`s hyperbolic space-time, we proceed as follows The required modified Schwarzschild spherically symmetric metric will be

GU HO GW HM GU U G8 GU O O GW M M GU U G8 for which the Schwarzschild metric is just an approximation

ϲϴ

GU HO GW HM GU U G8 GU x O GW M GU U G8 The Ricci tensor

5WW HOM O aa O a O aM a O a U L

From

5UU O aa O a O aM a M a U LL £¦ O a M a ¬²¦¦ 5RR ¦¤ HM U a HMU LLL ®»¦ ¦¥¦ ¼¦

5WW 5RR

web have

Write simply

O a M a , so O M NRQV WDQ W

M O ORJ N

Equation (i) is now just

HO U aa HO U B CU Equation (iii) is

HM U a a

H U N O

=C N We have now the complete solution ϲϵ

HM N NU x H N NU H N NU

O

H N B NU N N NU N N U

For radial motion, G8 .The Schwarzschild metric will be

GU HO GW HM GU

GU N N U GW H N NU GU

The free fall from rest of a star (of mass m and energy E) far from the center possesses8 ( N ¬ GW P U ® GU

GU ¬ N ¬ GW ® U ®

GU ¬ GU ¬ N N U H N NU GW ® GW ®

N U N N U H

N NU

GU ¬ GW ®

To our purpose for the hyperbolic spacetime, the velocity far away from the center would be 9d N D and consequently N N D

N U N U N U N D H

ϳϬ

N ¢ N D U ¯±

9

neglect the term N U and rearrange

¯

N U N U N D H N ¢ N D U±9 H

N ¢ N D U ¯±

9 N U N D

9H

N ¢ N D U ¯±

9H

DN ¢ DN U ¯±

N U N D N U N D

D N

= DN x D 9 H N U N U N D

6.4. The modified Schwarzschild metric in the hyperbolic spacetime for radial null trajectory is

GU N U N D GW H N U GU U G8 N U N D GW H N U GU GU N U H N U N D GW doesn`t possess singularity at U , since

ϳϭ

GU OLP HN U N U N D U l GW U l N U N U N D N U N D OLP OLP N U U l U l N HN U H U OLP U l N U N D H N U OLP

Note that Schwarzschild metric in the flat spacetime possesses singularity at U , since

GU N U GW N U GU U G8

for radial null trajectory is

GU N U GW N U GU

GU ¬ N U

GW ® GU U lR 9 N U ¶¶¶ ld GW

ϳϮ

6.5. Nonsingular Kerr Rotating Black Hole in the hyperbolic spacetime Kerr metric of a rotating black hole is given by9

NU ¬ NBU VLQ R S GU GW GGGW GU S GR % S ® S

NUB VLQ R ¬ U B VLQ RGG S ® Bw

N

S U B FRV R % U B NU The singularity arises when

S U B FRV R

Kerr metric reduces to Schwarzschild metric when

B

.

By analogy we can rewrite Kerr metric in the hyperbolic spacetime that can be reduced to the modified Schwarzschild metric in the hyperbolic spacetime, as follows:

S U B FRV R

% U B NU

ϳϯ

S U B NU ¬ % U B FRV R ®

S U B FRV R B FRV R B NU ¬ % U B FRV R ®

S B FRV R B NU ¬ U B FRV R % ® S B FRV R B NU ¬ U B FRV R % ® S H %

B FRV RB NU U B FRV R

S H %


Bl ¶¶¶ lH

N U

The Hyperbolic spacetime Kerr metric can be rewritten as B NU N ¬ NBU VLQ R GU GW GGGW H D® S S

FRV RB NU U B FRV R

GU S GR

NUB VLQ R ¬ U B VLQ RGG S ® which reduces to our modified Schwarzschild metric in the Hyperbolic spacetime when

B ϳϰ

GU N U N D GW H N U GU U G8 The Kerr metric in the hyperbolic spacetime for radial null trajectory doesn`t possess singularity at U , since

NU N¬ GU GW H U B FRV R D ®

NU N¬ GW H U B FRV R D ® B FRV RB NU U B FRV R

GU H GW



NU N U B FRV R D

S l º R Q DQG U l B NU U

GU H GW

OLP U l

NU N U D

GU GW

7KHOLPLWLVWDNHQE\/C+RVSLWDOCVUXOHZKHUHș ʌ

ϳϱ

GU

GU

7. Information loss paradox resolved by nonsingular hyperbolic spacetime10 Black holes owe their existence to the presence of singularity. Singularity appears theoretically as a result to the Schwarzschild solution in asymptotically flat spacetime. Such an approximated Schwarzschild solution creates singularity (when r = 0). This false paradigm constitutes our observation. The observer is operating within a "paradigm". Observations being made are not complete in themselves; they interpreted within a theory (a paradigm). Schwarzschild solution singularity paradigm works as a lunette, through which we imagine that we could observe Black holes. Black holes have never been seen directly, their existence is just a matter of illusion. We did prove that the spacetime of the actual Universe is hyperbolic [S. A. Mabkhout, Phys. Essays 25, 112. 2012)]. Neither Schwarzschild metric nor Kerr metric possess singularity in the hyperbolic spacetime [S. A. Mabkhout, Phys. Essays 26, 422. 2013)] . Singularity is the main character of the Black hole. If, in principle, singularity theoretically doesn't exist, Black holes also don`t exist. There is no singularity to crush and destruct the infalling information. In the actually hyperbolic spacetime infalling particles (information) have just come to rest at the origin (r = 0). Hence Information Loss Paradox does no longer exist. The principle of equivalence, which says that gravity couples to the energy-momentum

tensor

of

matter

and

the

quantum-mechanical

requirement that energy should be positive imply that gravity is always attractive. This leads to singularities in any reasonable theory of gravitation. A singularity is a place where the classical concepts of space and time break ϳϲ

down as do all the known laws of physics because they are all formulated on a classical spacetime background. The equivalence principle asserts that free-fall should feel the same as floating in empty space. Quantum Theory is the mathematics that is currently believed to underlie all physical proFHVVHV LQ QDWXUH ,W FDQ¶W EH XVHG WR SUHGLFW SUHFLVHO\ ZKDW will happen, but only the probability for any particular thing to happen. But probabilities only make sense if, when you add up all the probabilities for all of the different things that can possibly happen, you find the sum is equal to RQH $ TXDQWXP WKHRU\ ZKHUH WKLV LVQ¶W WUXH PDNHV QR VHQVH 2QH consequence of this is that in a quantum theory, information is never truly lost, nor is it truly copied; at least in principle, you can always determine KRZ D V\VWHP VWDUWHG LWV ³LQLWLDO VWDWH´ IURP FRPSOHWH LQIRUPDWLRQ DERXW KRZLWHQGVLWV³ILQDOVWDWH´ 11. According to the standard rules of quantum field theory in a fixed Minkowski spacetime, the time evolution of any system from a given initial state is described unambiguously by a unitary transformation acting on that state, and in this sense there is never any loss of fundamental information. 0RQRJDP\ LV D ULJRURXV UHVXOW RI TXDQWXP PHFKDQLFV GXEEHG µWKH PRQRJDP\RIHQWDQJOHPHQW¶VD\VWKDWRQH quantum system cannot be fully entangled with two independent systems at once. Monogamy stating that no particle can be entangled with two systems at the same time (while classical correlations can easily be shared by many parties, quantum correlations are harder to share). If Bob is highly entangled with Alice, that limits his ability WR HQWDQJOH ZLWK &DUULH DQG LI KH HQWDQJOHV ZLWK &DUULH LQVWHDG KH FDQ¶W HQWDQJOHZLWK$OLFH+HQFHZHVD\WKDWHQWDQJOHPHQWLV³PRQRJDPRXV Pure state is the quantum state where we have exact information about the ϳϳ

quantum system. A system is said to be in a pure state if we have complete knowledge about that system, meaning we know exactly which state it's in. The S-matrix is the unitary operator S that determines the evolution of the initial state to the final state. A system is in a mixed state if we only have partial (or no) knowledge of the system. In terms of a probability density, ȡ say, this means that more than one of its eigenvalues must be non-zero. The mixed state is the combination of probabilities of the information about the quantum state of the quantum system. A mixed state described by a density matrix. As you may recall, non-unitary evolution is not allowed to occur naturally in a quantum theory because it fails to preserve probability; that is, after non-unitary evolution, the sum of the probabilities of all possible outcomes of an experiment may be greater or less than one. Unitary evolution is reversible while non- unitary evolution is irreversible. Quantum theory is reversible. We did prove that the curvature of the Universe is negative (k=-1). Hence, the spacetime of the Universe is hyperbolic. 7.1. Hawking Radiation S. W. Hawking, an English theoretical physicist, was one of the first to consider the details of the behavior of a black hole whose Schwarzschild radius was on the level of an atom. These black holes are not necessarily low mass, for example, it requires 1 billion tons of matter to make a black hole the size of a proton. But their small size means that their behavior is a mix of quantum mechanics rather than relativity ³+RZHYHULWLVVKRZQWKDWTXDQWXP mechanical effects cause black holes to create and emit particles as if they were hot bodies "12. ϳϴ

Before black holes were discovered it was know that the collision of two photons can cause pair production. This direct example of converting energy into mass (unlike fission or fusion which turn mass into energy). Pair production is one of the primary methods of forming matter in the early Universe. Note that pair production is symmetric in that a matter and antimatter particle is produced (an electron and an anti-electron, positron). Hawking showed that the strong gravitational gradients (tides) near black holes can also lead to pair production. In this case, the gravitational energy of the black hole is converted into particles. "If the matter/anti-matter particle pair is produced below the event horizon, then particles remain trapped within the black hole. But, if the pair is produced above the event horizon, it is possible for one member to fall back into the black hole, the other to escape into space. Thus, the black hole can lose mass by a quantum mechanical process of pair production outside of the event horizon"13. The rate of pair production is stronger when the curvature of spacetime is high. Small black holes have high curvature, so the rate of pair production is inversely proportional to the mass of the black hole (this meDQVLW¶VIDVWHUIRU smaller black holes). Thus, Hawking was able to show that the mini or primordial black holes expected to form in the early Universe have since disintegrated, resolving the dilemma of where all such mini-black holes are today. In 1975 Hawking published a shocking result: if one takes quantum theory into account, it seems that black holes are not quite black! Instead, they should glow slightly with "Hawking radiation", consisting of photons, neutrinos, and to a lesser extent all sorts of massive particles. Virtual particle ϳϵ

pairs are constantly being created near the horizon of the black hole, as they are everywhere. Normally, they are created as a particle-antiparticle pair and they quickly annihilate each other. But near the horizon of a black hole, it's possible for one to fall in before the annihilation can happen, in which case the other one escapes as Hawking radiation. This has never been observed, since the only black holes we have evidence for are those with lots of hot gas falling into them, whose radiation would completely swamp this tiny effect. :H ZRQ¶W VHH DQ\ RI WKH EODFN KROHV LQ WKH 0LON\ :D\ H[SORGH DQ\ WLPH soon though, not only are they likely still gaining mass (from the cosmic microwave background, at least), but a one sol black hole would take over 10^67 years to evaporate (the universe is only 13 billion years old)! 7.2. Information Loss Paradox The black hole creates particles in pairs, with one particle always falling into the hole and the other possibly escaping to infinity. Because part of the information about the state of the system is lost down the hole, the final situation is represented by a density matrix rather than a pure quantum state. This means there is no S matrix for the process of black-hole formation and evaporation. Instead one has to introduce a new operator, called the superscattering operator, which maps density matrices describing the initial situation to density matrices describing the final situatiRQ +DZNLQJ¶V argument basically comes down to the observation that in the quantum realm, µHPSW\¶ VSDFH LVQ¶W HPSW\ 'RZQ DWWKLVVXE-sub-microscopic level, it is in constant turmoil, with pairs of particles and their corresponding antiparticles continually popping into existence before rapidly recombining and vanishing. Only in very delicate laboratory experiments does this submicroscopic ϴϬ

frenzy have any observable consequences. But when a particle±antiparticle SDLUDSSHDUVMXVWRXWVLGHDEODFNKROH¶VHYHQt horizon, Hawking realized, one member could fall in before the two recombined, leaving the surviving partner to fly outwards as radiation. The doomed particle would balance the positive energy of the outgoing particle by carrying negative energy inwards ² something allowed by quantum rules. That negative energy would then JHW VXEWUDFWHG IURP WKH EODFN KROH¶V PDVV FDXVLQJ WKH KROH WR VKULQN %XW with it came the disturbing realization that black-hole radiation leads to a paradox that challenges quantum theory. ,Q KLV DUWLFOH ³%UHDNGRZQ RI 3UHGLFWDELOLW\ LQ *UDYLWDWLRQDO &ROODSVH´ 14 Stephen Hawking argues that his prediction that black holes emit thermal radiation implies that the evolution of black holes cannot be described by standard unitary quantum mechanical evolution. This nonunitary evolution is popularly described as representing a loss of ³LQIRUPDWLRQ´± if a pure state nonunitarily evolves into a mixture, then we can no longer predict with certainty the outcome of any complete set of measurements, thus it appears that some previously existing information has been destroyed. This conclusion has been generally viewed as unacceptable by high energy physicists ± ZKR KDYH WKHUHIRUH FKDUDFWHUL]HG +DZNLQJ¶V DUJXPHQWDVD³SDUDGR[WKDWQHHGVWREHUesolved" 15. In principle, it should be possible to recover everything there is to know about the objects that fell in a black hole by measuring the quantum state of the radiation coming out. But Hawking showed that it was not that simple: the radiation coming out is random. Toss in a kilogram of rock or a kilogram of computer chips and the result will be the same. Watch the black hole even until it dies, and there would still be no way to tell how it was formed or what fell in it. The black ϴϭ

hole is gone. Where did the information go? If it disappeared along with the black hole, that violates quantum theory. "Maybe the information came back out with the Hawking radiation? The problem is that the information in the EODFNKROHFDQ¶WJHWRXW6RWKHRQO\ZD\LW can be in the Hawking radiation (naively) as if what is inside is copied. Having two copies of the information, one inside, one outside, also violates quantum theory" 11. This problem, dubbed the black-hole information paradox, divided physicists into two camps. Some, like Hawking, argued that the information truly vanishes when the black hole dies. If that contradicted quantum laws, then better laws needed to be found. Of course, it may simply be that quantum theory is incomplete, and that the physics of black holes forces us to extend that theory. And this is what Hawking believed for three decades. "In order to understand why the information loss problem is a problem, we need first to understand what it is. Take a quantum system in a pure state and throw it into a black hole. Wait for some amount of time until the hole has evaporated enough to return to its mass previous to throwing anything in. What we start with is a pure state and a black hole of mass M. What we end up with is a thermal state and a black hole of mass M. We have found a process (apparently) that converts a pure state into a thermal state. But, and here's the kicker, a thermal state is a MIXED state (described quantum mechanically by a density matrix rather than a wave function). In transforming between a mixed state and a pure state, one must throw away information" 16. As you may recall, non-unitary evolution is not allowed to occur naturally in a quantum theory because it fails to preserve probability; that is, after non-unitary evolution, the sum of the probabilities of all possible outcomes of an experiment may be greater or less than one. ϴϮ

7.3. Black Hole Complementarity (An Attempt to Resolve the Paradox) Leonard Susskind

7

proposed a radical resolution to this problem by

claiming that the information is both reflected at the event horizon and passes through the event horizon and can't escape, with the catch being no observer can confirm both stories simultaneously. According to an external observer, the infinite time dilation at the horizon itself makes it appear as if it takes an infinite amount of time to reach the horizon. He also postulated a stretched horizon, which is a membrane hovering about a Planck length outside the event horizon and which is both physical and hot. According to the external observer, infalling information heats up the stretched horizon, which then reradiates it as Hawking radiation, with the entire evolution being unitary. However, according to an infalling observer, nothing special happens at the event horizon itself, and both the observer and the information will hit the singularity. This isn't to say there are two copies of the information lying about ² one at or just outside the horizon, and the other inside the black hole ² as that would violate the no cloning theorem. Instead, an observer can only detect the information at the horizon itself, or inside, but never both simultaneously. Complementarity is a feature of the quantum mechanics of noncommuting observables, and Susskind proposed that both stories are complementary in the quantum sense. Complementarity claims that an outside observer can effectively described the black hole as a heated membrane situated just above the event horizon. According to the outside observer, any infalling information will interact violently with this heated membrane, and will eventually be reemitted to exterior universe, thus keeping the late-time state pure. A difficulty facing this suggestion is the fact that the event horizon of a black ϴϯ

hole is a globally defined property of a spacetime; we would not expect a freely falling observer to notice anything unusual at the horizon ± and we certainly would not expect her to be destroyed there. Black hole complementarity postulates that from her perspective WKHLQIDOOLQJREVHUYHU¶V passage through the heated membrane and the event horizon is indeed uneventful. Her description of the situation is claimed to be complementary WRWKHH[WHUQDOREVHUYHU¶VGHVFULSWLRQUDWKHULQWKHZD\WKDWWKHGHVFULSWLRQV of a quantum particle in terms of position and momentum are complementary. Hawking had shown that the quantum state of any one particle escaping from the black hole is random, so the particle cannot be carrying any useful information. But in the mid-1990s, Susskind and others "realized that information could be encoded in the quantum state of the radiation as a ZKROH LI WKH SDUWLFOHV FRXOG VRPHKRZ KDYH WKHLU VWDWHV µHQWDQJOHG¶ ² intertwined in such a way that measurements carried out on one will immediately influence its partner, no matter how far apart they are" 17. * ¶W +RRIW

18

proposed, an approach to black hole quantization is

proposed wherein it is assumed that quantum coherence is preserved. Pure states could evolve into mixed states, as the thermal character of Hawking radiation has been taken to indicate. At first sight it might seem that the question of whether quantum coherence gets lost has little to do with physics on Planckian energy scales. The original derivation by Hawking that the expectation values of all operators as experienced by late observers are described by mixed quantum states, seemed to be totally independent of Planck scale details. Yet, the argument did involve the spacetime geometry DUELWUDULO\ ³FORVH´ WR WKH FODVVLFDOO\ GHWHUPLQHG KRUL]RQ DQG LQFOXGHG ϴϰ

energies for which the gravitational redshift had become arbitrarily large. Moreover, the fact that the outgoing particles look thermal will be affected by any interactions occurring very near the horizon and, in turn, these might even reconvert apparently mixed states back into pure states in such a way that an outside observer could hardly tell the difference, any more easily than he could for a bucket of water. Does Complementarity Saving Quantum Theory? However, others felt that it was general relativity, not quantum theory, that would need to be FKDQJHG $QG D SURSRVDO ZDV PDGH LQ FDOOHG ³FRPSOHPHQWDULW\´ suggested that the information was, in a sense, both inside and outside but without violating quantum theory. (This proposal was developed by Susskind). Specifically, observers who remain outside the black hole see the information accumulate at the horizon, and then come flying outward in the Hawking radiation. Observers who fall into the black hole see the information located inside. Since the two classes of observers cannot communicate, there is no paradox. Still, this suggestion is potentially selfcontradictory, and requires a number of strange things be true. Among them is something called ³KRORJUDSK\´ DQ LGHD GHYHORSHG E\ * µW +RRIW DQG further by Susskind. The idea is that the physics of the three-dimensional interior of the black hole, where gravity obviously plays a role, can instead be viewed, via a rather mysterious transformation, as physics just above the two-dimensional horizon, where it is described by two-dimensional equations that do not include gravity at all. "Crazy at it sounds, considerable evidence arose in the late 1990s that it is true, at least in some situations! In 1997, Maldacena conjectured that under the right circumstances, string theory is actually equivalent to a quantum ϴϱ

WKHRU\ VSHFLILFDOO\ D ³quantum field theory´ ZLWKRXW JUDYLW\ DQG ZLWK fewer dimensions. 7KLVUHODWLRQVKLSNQRZQYDULRXVO\DV³$G6&)7´RUWKH ³ILHOGVWULQJ´FRUUHVSRQGHQFHGHVHUYHVDQDUWLFOHDOOLWVRZQ " 1. The success of holography gave additional credence to the complementarity idea. Furthermore, the field/string correspondence allowed for a very strong argument that small black holes can form and evaporate in the string theory via a process that can be described by the corresponding quantum field theory (though not explicitly) ² and which therefore, as in all processes in any quantum theory, does preserve information! By 2005, even Hawking had come around to this point of view ² that in fact, as the complementarity proposal had suggested, black holes do not cause information to be lost, and that general relativity, but not quantum theory, must be modified. Still, there were loose ends in the complementarity proposal. Black hole evaporation is so subtle that there were still no quantum theory equations for complementarity that could describe the evaporation process. 7.4. Firewall (failure of Complementarity) Black hole evaporation is so subtle that there were still no quantum theory equations for complementarity that could describe the evaporation process. While trying to find such equations, Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully

19

discovered that in fact (at least under

reasonable assumptions) complementarity contains a self-contradiction, which shows up when a black hole has evaporated about halfway. The DUJXPHQWLVH[WUHPHO\VXEWOHLQYROYLQJWKHNLQGRI³TXDQWXPHQWDQJOHPHQW´ But crudely speaking, by the halfway point, so much information has GHSDUWHGWKHEODFNKROHLQWKH+DZNLQJUDGLDWLRQWKDWWKHUH¶VQRWHQRXJKOHIW ϴϲ

at the horizon for holography WR UHSUHVHQW WKH EODFN KROH¶V LQWHULRU Consequently, instead of an in-falling observer smoothly entering the black KROHWKURXJKWKHKDUPOHVV KRUL]RQ WKHREVHUYHU ILQGV WKHUH¶VQRLQWHULRU DW all, and does so the hard way, by being fried to a crisp by a so-called ³firewall´ WKDW KRYHUV MXVW RXWVLGH WKH KRUL]RQ1. In their account, quantum effects would turn the event horizon into a seething maelstrom of particles. Anyone who fell into it would hit a wall of fire and be burned to a crisp in an instant. Such firewalls would violate a foundational tenet of physics that was first articulated almost a century ago by Albert Einstein, who used it as the basis of general relativity, his theory of gravity. Known as the equivalence principle, it states in part that an observer falling in a gravitational field ² even the powerful one inside a black hole ² will see exactly the same phenomena as an observer floating in empty space. Without this principle, (LQVWHLQ¶V IUDPHZRUN FUXPEOHV :HOO DZDUH RI WKH LPSOLFDWLRQV RI WKHLU claim, Polchinski and his co-authors offered an alternative plot ending in which a firewall does not form. But this solution came with a huge price. Physicists would have to sacrifice the other great pillar of their science: quantum mechanics, the theory governing the interactions between subatomic particles. Hawking had shown that the quantum state of any one particle escaping from the black hole is random, so the particle cannot be carrying any useful information. But in the mid-1990s, Susskind and others realized that information could be encoded in the quantum state of the radiation as a whole iI WKH SDUWLFOHV FRXOG VRPHKRZ KDYH WKHLU VWDWHV µHQWDQJOHG¶ ² intertwined in such a way that measurements carried out on one will ϴϳ

immediately influence its partner, no matter how far apart they are. But how FRXOGWKDWEHZRQGHUHGWKH3ROFKLQVNL¶VWHDP")Rr a particle to be emitted at all, it has to be entangled with the twin that is sacrificed to the black hole. And if Susskind and others were right, it also had to be entangled with all the Hawking radiation emitted before it. Yet a rigorous result of quantum PHFKDQLFVGXEEHGµWKHPRQRJDP\RIHQWDQJOHPHQW¶VD\VWKDWRQHTXDQWXP system cannot be fully entangled with two independent systems at once. To escape this paradox, Polchinski and his co-workers realized, one of the entanglement relationships had to be severed. Reluctant to abandon the one required to encode information in the Hawking radiation, they decided to snip the link binding an escaping Hawking particle to its infalling twin. But WKHUH ZDV D FRVW ³,W¶V D YLROHQW SURFHVV OLNH EUHDNLQJ WKH ERQGV of a PROHFXOHDQGLWUHOHDVHVHQHUJ\´VD\V3ROFKLQVNL7KHHQHUJ\JHQHUDWHGE\ VHYHULQJ ORWV RI WZLQV ZRXOG EH HQRUPRXV ³7KH HYHQW KRUL]RQ ZRXOG OLWHUDOO\ EH D ULQJ RI ILUH WKDW EXUQV DQ\RQH IDOOLQJ WKURXJK´ KH VD\V $QG that, in turn, violates the equivalence principle and its assertion that free-fall should feel the same as floating in empty space ² impossible when the former ends in incineration. So they posted a paper on the preprint server, arXiv, presenting physicists with a stark choice: either accept that firewalls exist and that general relativity breaks down, or accept that information is lost in black holes and quantum mechanics is wrong. Firewalls seem like the least crazy option. The paper rocked the physics community. It was outrageous to FODLP WKDW JLYLQJ XS (LQVWHLQ¶V HTXLYDOHQFH SULQFLSOH LV WKH best option. Raphael %RXVVRVD\V³$ILUHZDOOVLPSO\FDQ¶WDSSHDULQHPSW\ space; any more than a brick wall can suddenly appear in an empty field and VPDFN \RX LQ WKH IDFH´ ,I (LQVWHLQ¶V WKHRU\ GRHVQ¶W DSSO\ DW WKH HYHQW ϴϴ

horizon, cosmologists would have to question whether it fully applies anywhere. "So now the paradox is back! And worse than ever. It seems that if TXDQWXP WKHRU\ DQG FRPSOHPHQWDULW\ DUH ULJKW JHQHUDO UHODWLYLW\ LVQ¶W MXVW UHTXLULQJVRPHVPDOOPRGLILFDWLRQLWUHTXLUHVPDMRUVXUJHU\$QGWKHUH¶VQR sign of such surgery in string theory, which provided the example of holography. But the field/string correspondence suggests quantum theory can describe black hole formation and HYDSRUDWLRQVRLQIRUPDWLRQLVQ¶WORVW 1. 7.5. S. W. +DZNLQJ(YHQW+RUL]RQ'RHVQ¶W([LVW 0RVW SK\VLFLVWV IRROKDUG\ HQRXJK WR ZULWH D SDSHU FODLPLQJ WKDW ³WKHUH DUHQREODFNKROHV´² at least not in the sense we usually imagine ² would probably be dismissed as cranks. But when the call to redefine these cosmic FUXQFKHUVFRPHVIURP6WHSKHQ+DZNLQJLW¶VZRUWKWDNLQJQRWLFH,QDSDSHU posted online, the physicist, based at the University of Cambridge, UK, and one of the creators of modern black-hole theory, does away with the notion of an event horizon, the invisible boundary thought to shroud every black KROHEH\RQG ZKLFKQRWKLQJQRWHYHQOLJKW FDQHVFDSH +DZNLQJ¶V UDGLFDO SURSRVDOLVDPXFKPRUHEHQLJQ³DSSDUHQWKRUL]RQ´ZKLFKRQO\WHPSRUDULO\ holds matter and energy prisoner before eventually releasing them, albeit in a more garbled form. There is no escape from a black hole in classical theory. 4XDQWXPWKHRU\KRZHYHU³HQDEOHVHQHUJ\DQGLQIRUPDWLRQWRHVFDSHIURPD EODFN KROH´ $ IXOO H[SODQDWLRQ Rf the process, the physicist admits, would require a theory that successfully merges gravity with the other fundamental forces of nature. But that is a goal that has eluded physicists for nearly a FHQWXU\³7KHFRUUHFWWUHDWPHQW´+DZNLQJVD\V³UHPDLQVDP\VWHU\´20 ϴϵ

Hawking posted his paper on the arXiv preprint. He titled it, whimsically, 'Information preservation and weather forecasting for black holes'

21

, and it

has yet to pass peer review. The paper was based on a talk he gave via Skype at a meeting at the Kavli Institute for Theoretical Physics in Santa Barbara, California, in August 2013. Hawking's new work is an attempt to solve what is known as the black-hole firewall paradox, which has been vexing physicists for almost two years. Now Hawking proposes a third, tantalizingly simple, option. Quantum mechanics and general relativity remain intact, but black holes simply do not have an event horizon to catch fire. The key to his claim is that quantum effects around the black hole cause space-time to fluctuate too wildly for a sharp boundary surface to exist. In SODFH RI WKH HYHQW KRUL]RQ +DZNLQJ LQYRNHV DQ ³apparent horizon´ D surface along which light rD\VDWWHPSWLQJWRUXVKDZD\IURPWKHEODFNKROH¶V core will be suspended. In general relativity, for an unchanging black hole, these two horizons are identical, because light trying to escape from inside a black hole can reach only as far as the event horizon and will be held there, as though stuck on a treadmill. However, the two horizons can, in principle, be distinguished. If more matter gets swallowed by the black hole, its event horizon will swell and grow larger than the apparent horizon. Conversely, in the 1970s, Hawking also showed that black holes can slowly shrink, spewing out 'Hawking radiation'. In that case, the event horizon would, in theory, EHFRPHVPDOOHUWKDQWKHDSSDUHQWKRUL]RQ+DZNLQJ¶VQHZVXJJHVWLRQLVWKDW the apparent horizon is the UHDO ERXQGDU\ ³7KH DEVHQFH RI HYHQW KRUL]RQV means that there are no black holes ² in the sense of regimes from which OLJKWFDQ WHVFDSHWRLQILQLW\´+DZNLQJZULWHV+DZNLQJ VDWWHPSWWRUHVROYH the paradox is criticized by: ϵϬ

Don Page 20, a physicist and expert on black holes at the University of Alberta in Edmonton, Canada, who collaborated with Hawking in the 1970s, VD\V³7KHSLFWXUH+DZNLQJJLYHVVRXQGVUHDVRQDEOH´³SDSHU@ LV QRQVHQVH´ 6KH PLVXQGHUVWDQG +DZNLQJ radiation, and assume that matter behaves in ways that are completely LPSODXVLEOH´ he claimed. Quite to the contrary of Mersini-Houghton calculations, Unruh maintains that black holes do not emit enough Hawking radiation to lose mass to avoid formation of a black hole"17. Unruh explained, adding that it is a common mistake for those who do not understand +DZNLQJ¶V UDGLDWLRQ WKHRU\ LQ IXOO WKDW WKH ³RXWJRLQJ HQHUJ\ EDFN FORVHU and closer to the horizon of the black hole, where its energy density gets ODUJHU DQG ODUJHU´ he said. ³8QIRUWXQDWHO\ H[SOLFLW FDOFXODWLRQV RI WKH ϭϬϯ

energy density near the horizon show it is really, really small instead of EHLQJODUJH³7KH>SDSHU@LVQRQVHQVH´ In flat spacetime Schwarzschild metric (Kerr metric) possesses singularity (when r=0 , the velocity is infinite). We did prove mathematically that the spacetime of the Universe is hyperbolic. We show that neither the modified Schwarzschild metric nor the modified Kerr metric possess singularity in the hyperbolic spacetime. Singularity is the main character of the Black hole. If, in principle, singularity theoretically doesn't exist, Black holes also don`t exist. There is no singularity to crush and destruct the infalling information. In the actually hyperbolic spacetime infalling particles (information) have just come to rest at the origin (r = 0). Hence Information Paradox does no longer exist.

ϭϬϰ

summary 1. The Big Bang`s Hyperbolic Universe possesses no singularity. 2. The Schwarzschild hyperbolic spacetime possesses no singularity. 3. The Hyperbolic Universe grows exponentially preserve a legitimate inflation covers the current large structure. 4. The state W M S M 5 M q \U JP FP FP is consistent with current observable universe. 5.The Big Bang provide a consistent initial condition, since the hyperbolic equation (5) governs the time evolution of the Universe since the Big Bang through discrete states W M S M 5 M and successfully path through Planck epoch. 6. Information loss paradox resolved by nonsingular hyperbolic spacetime

ϭϬϱ

References 1

2

http://en.wikipedia.org/wiki/Dark_energy. (access day 8/9/2012)

http://star-

www.stand.ac.uk/~spd3/Teaching/AS1001/AS1001_10.pdf(accessday 1/9/2012) 3

http://en.wikipedia.org/wiki/Hyperbolic_trajectory (access day 1/9/2012)

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The official string theory web site cosmology

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Nature 510, 9 (05 June 2014) doi:10.1038/510009a

http://www.nature.com/news/big-bang-blunder-bursts-the-multiversebubble-1.15346 6

arxiv.org/abs/1409.6530

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http://universe-review.ca/F15- 15particle.htm

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Edwin F.Tylor. John A. Wheeler,(2000),Exploring Black Holes

introduction to General relativity, Addison Wesley Longman.p 3-12. 9

James B. Hartle(2003): Gravity An Introduction To Einstein's General

Relativity. Addison Wesley. P 311 10

Salah A. Mabkhout, Information Loss Paradox Resolved by Nonsingular

Hyperbolic Spacetime, International Journal of Astrophysics and Space Science. Vol. 2, No. 5, 2014, pp. 71-80. doi: 11

Matt Strassler, Of Particular Significance Conversations About Science

profmattstrassler.com/ 12

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199-220 Springer- Verlag (1975) . 13

http://abyss.uoregon.edu/~js/ast123/lectures/lec17.html

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S. W. Hawking, Breakdown of predictability in gravitational collapsePhys.

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cv.pdf 16

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http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/info_loss.html 17

Leonard Susskind, L´arus Thorlacius, and John Uglum, The Stretched

Horizon and Black Hole Complementarity arXiv:hep-th/9306069v2 28 Jun 1993 18

& 5 6WHSKHQV * ¶W +RRIW and B. F. Whiting, BLACK HOLE

EVAPORATION

WITHOUT

INFORMATION

LOSS.

arXiv:gr-

qc/9310006v1 4 Oct 1993 19

Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully,

Black Holes: Complementarity or Firewalls? arXiv:1207.3123v4 [hep-th] 13 Apr 2013. 20Nature 496, 20±23 (04 April 2013) doi:10.1038/496020a, Notion of an 'event horizon', from which nothing can escape, is incompatible with quantum theory, physicist claims. 21

Stephen Hawking, 'There are no black holes' : arXiv:1401.5761v1 [hep-th] 22

Jan 2014 ϭϬϳ

22

Nature doi:10.1038/nature.2014.14583

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REBECCA JACOBSON , :KDW+DZNLQJPHDQWZKHQKHVDLGµWKHUHDUH

no black KROHV¶http://www.pbs.org/newshour/updates/hawking-meant-blackholes/ 24

http://www.creationstudies.org/Education/big_bang.html

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S. Hossenfelder and L. Smolin, ,Restoring unitary evolution relies on

elimination of singularities. (arXiv:0901.3156) 26

Laura Mersini-Houghton, Black holes do not exist! Backreaction of

Hawking Radiation on a Gravitationally Collapsing Star I Black Holes? arXiv:1406.1525v1 [hep-th] 5 Jun 2014. 27

http://rt.com/news/190540-black-holes-are-nonexistent/

28

Salah. A. Mabkhout (2012), The hyperbolic geometry of the universe and

the wedding of general relativity theory to quantum theory. Physics Essays: March 2012, Vol. 25, No. 1, pp. 112-118. 29

Salah. A. Mabkhout (2013) The Big Bang hyperbolic universe neither

needs inflation nor dark matter and dark energy. Physics Essays: September 2013, Vol. 26, No. 3, pp. 422-429. 30

31

Roger Penrose (2004): THE ROAD TO REALITY, Jonathan Cape, p 1023. Steven B. Giddings Black holes, quantum information, and the foundations

of physics Citation: Physics Today 66(4), 30 (2013); doi: 10.1063/PT.3.1946

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Figure 1 http://www.astro.cornell.edu/academics/courses/astro201/geometry.htm

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Part II

Modification of the laws of gravity based on the hyperbolic spacetime resolved the mystery of both dark energy and dark matter and explains the superluminous speed of distant objects.

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1. Introduction The 1990's two teams of astronomers, the Supernova Cosmology Project (Lawrence Berkeley National Laboratory) and the High-Z Supernova Search (international) were looking for distant Type Ia supernovae in order to measure the expansion rate of the Universe with time. They expected that the expansion would be slowing, which would be indicated by the supernovae being brighter than their redshifts would indicate. Instead, they found the supernovae to be fainter than expected from a uniformly expanding universe. Hence, the expansion of the Universe was accelerating! In the flat Universe where the curvature is zero and the density is the critical density, new hypothetical objects, dark energy and dark matter, are essential to bridge the gap between theory and observation. It is clear that these new ingredients, dark energy and dark matter, arose as a consequence to the false flat universe paradigm. Without them, there would be a fatal contradiction between the observations made by astronomers and the predictions of the big bang theory. We modify the laws of gravity based on a hyperbolic spacetime rather than a flat spacetime [S. A. Mabkhout, Phys. Essays 26,422 (2013)]. Such a modification explains the accelerating expansion of the Universe without need for dark energy and also explains the flat rotation curve without need for dark matter. Moreover , recession velocities exceed the speed of light in all viable cosmological models for objects with red shifts greater than z ~ 1.5. Locally, the spacetime is flat where the speed of light is the constant "c". The hyperbolic structure of the spacetime could only be UHFRJQL]DEOHJOREDOO\)DVWHUWKDQOLJKWPRWLRQRFFXUVRXWVLGHWKHREVHUYHU¶V inertial frame (local frame). As the objects are far distant separated as much ϭϭϮ

the spacetime is hyperbolic curved, as faster the velocity of light exceeds the constant speed "c" of light due to flat spacetime. The flat spacetime constraints the domain of applicability of special relativity. Cosmology, the study of the universe at its largest scales, is undergoing a period of intense development based on both rigorous theoretical and H[SHULPHQWDOLQSXW7KHGRPLQDQWIRUFHLQFRVPRORJ\LVJUDYLW\(LQVWHLQ¶V theory of General Relativity (GR), our modern theory of gravity, forms a solid theoretical underpinning of the recent progress. On the experimental side the discovery by Hubble of the expanding universe is the basis of the Big Bang model of a dynamic universe that allows a meaningful discussion of a beginning, and thus an age of the universe. The discovery of the Cosmic Microwave Background (CMB), the relic sea of photons left over from the early universe, further supports the Big Bang model. The realization from the rotation curves of galaxies that there is considerably more matter with attractive gravity, called dark matter, than we can see via electromagnetic radiation (the visible matter) complicates things. With attractive gravity between the components of the universe the expectation was that the expansion velocity of the universe, the so called Hubble velocity, would decrease with time, i.e. a decelerating universe. The recent discovery that the Hubble velocity is increasing with time, an accelerating universe, was an immense surprise. This led to the postulation of the existence of a new component of the universe, dark energy, with some very unusual properties, not the least among which is that it has, in some sense of the word, repulsive gravity that drives the acceleration. The best estimates that fit all of the presently existing data is that the energy/matter density of the universe is ϭϭϯ

made of, 4% of visible matter, dark matter making up 24% and dark energy making up the majority 72%. The Type 1a supernova as standard candles can be used to measure the expansion history of the universe i.e. the plot of the scale factor of the universe when the supernova light was emitted versus the time in the past when the supernova explosion occurred. This is done by measuring the apparent brightness L of the supernova and its redshift z. Comparing the apparent brightness to the presumably known intrinsic brightness of the supernova determines its distance and from the distance the time in the past when the supernova exploded can be inferred knowing the velocity of light. The redshift gives the scale factor of the universe at the time of the supernova explosion via a(t) = 1/(1+z). Each supernova then yields a point in the a(t) versus t plot, and a large sample of supernovae thus measure the expansion history of the universe. Given the high level of interest in understanding the underlying reasons for the acceleration of the universe and the nature of dark energy, a vigorous program is planned worldwide to measure the cosmological parameters to a higher level of precision. In particular there is a focus on the measurement of the equation of state parameter w, which can distinguish between a cosmological constant (w= í 1) or some other form of dark energy with w < í1/3 but not exactly í1. The present dependence of w, which would be inconsistent with a cosmological constant , is very poorly constrained by the present data. It is results, w = í0.93±0.16 is consistent with the cosmological constant but higher precision may yet show a deviation from í1. The parameter wa, indicating a time interesting to point out that if it turns out that dark energy LVGXHWRDQRQ]HURFRVPRORJLFDOFRQVWDQWDYDOXHRI ȍ ȁ in the vicinity of 0.7 would be implied. 6XFK D VPDOO YDOXH RI ȍȁ would lead to one of the ϭϭϰ

most dramatic inconsistencies in modern physics called the cosmological constant problem[1]. Modern field theory of particle physics predicts that the energy density of the vacuum contributes to the cosmological constant with a value 120 orders of magnitude larger than the experimentally observed value. Actually this problem would persist even if the dark energy turned out WR EH VRPHWKLQJ HOVH WKDQ WKH FRVPRORJLFDO FRQVWDQW ZLWK ȁ (YHQ though many theorists have tried, there is no explanation in sight to this fundamental disagreement. Accelerating, ruling out a nonaccelerating universe and a zero cosmological constant with a very high level of confident. All of the surveys are consistent with w = í1, assuming a constant w and a flat universe. We exhibit the hyperbolic structure of the universe that explains the accelerating expansion of the universe without need for an additional components, dark energy and dark matter.

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2 Method of solution of the dynamical equations in the presence of the cosmological constant The shape of the universe depends on the curvature k (the field strength) according to the following three cases (1) If K=0 then the universe is flat and open. (2) If K>0 then the universe is spherical and closed. (3) If K