The sphere surface at which the escape velocity equals c is called the event horizon, ... This literally means that at the event horizon the light can't escape the ...
Nothing can ever fall into blackholes (at least not till the end of time) by Eliezer M. Jacob Abstract: It is common knowledge blackholes swallows everything around them... I will try to prove that this assumption is inaccurate. To who this paper is intended: Anyone that wants to know. If you are afraid of mathematics, jump over the equations and read only the explanations. If you have good physics knowledge, jump over the explanation paragraphs and get directly get into the issues. What do we know or think we know: It is common to hear all around that when something is cached by any blackhole gravity it will inevitably fall into the blackhole, first passing trough the event horizon disappearing from our sensors/vision/detection, then finally colliding with the singularity that resides in the center of the blackhole. This is the explanation for the public hysteria when the LHC was turned on from the first time: “A micro-blackhole could be generated by the LHC, immediately starting to swallow all around it, including all Earth and later even the Sun...” In the more scientific community it is a consensus that a falling object into the blackhole will don't even sense it is passing trough the event horizon, however we will see him slowing down. It is not denied that the object falling down into the black hole will eventually pass trough the event horizon and eventually (but inevitably) hit the singularity. I pretend to prof here, this is not completely true, and till the end of time, nothing is capable to pass trough the event horizon of any wormhole, despite its size, is not important if it is a micro-blackhole generated by the LHC or a huge hyper-massive blackhole like the one on the center of our Milky Way. Conceptually: A blackhole is called that because even the light velocity1 doesn’t reach the escape velocity from the blackhole (i.e. can't escape from his gravity). The sphere surface at which the escape velocity equals c is called the event horizon, which has a radius as follows (called the Schwarzschild radius) : Being:
Where: Vesc G M R
: Escape velocity : Newton Gravitational constant : The mass : The distant from the center of mass (the singularity)
1 - Light velocity, I mean c the propagation velocity of waves in vacuum – meaning the fastest possible local velocity accordingly with Einstein Page 1/7
Setting :
Vesc = c in order to get the radius of the event horizon Result in : Where:
Rsw = Schwarzschild radius
This literally means that at the event horizon the light can't escape the blackhole, but also on the other side means that an object approaching the event horizon will “slow down” as it gets near the event horizon. Let me explain why in the next paragraphs. Space-time and light cones: We call space-time the 4 dimensions we commonly use, the time ( t) and the 3 space dimensions (X,Y,Z). Now we can plot the trajectory of a particle in that space time. Let for practical reasons plot all the 3 space dimensions (X,Y,Z) as one single coordinate S (space) and the time perpendicular to it. Also let set a convenient units system where c=1 , there a particle p traveling at any possible speed will be always confined within the dashed lines (see figure 1) that are at 45o . The trajectory done by the particle p is called the life-line and the angle represented by the dotted lines the light cone since if the particle travels at the speed of light its motion will be parallel to those doted lines
t
future
0
p'
p S Figure 1 : life-line and light cones in space-time p' represent te same particle p at a future time
Page 2/7
Back to blackholes Previously we find that at the event horizon the escape velocity equals the speed of light, lets therefore draw a graphical representation of this effect:
t p' future
s past
p
Here the particle p could pass trough the event horizon at p' since the dashed lines representing light speed now also represent the event horizon. Notice that now if at p' the particle regrets from crossing the event horizon will at most stay on it but it would never be able to get far from it even traveling in the opposite direction at maximal speed, the speed of light.
Figure 2 : Representation of a blackhole in space-time diagram space (also called an accelerated reference frame, or Minkowski diagram) Lets look a bit more in detail The dashed lines represented the event horizon, that is the spherical surface or radius: Rsw (Schwarchild radius) therefore the colored lines will represent regions further away from the blackhole
t Event horizon
r1 r2 r3 s
Blackhole
3a 3b Figure 3 : Parallel of an accelerated reference frame diagram with normal space blackhole representation Page 3/7
On the accelerated reference frame the surfaces of constant radius r1 r2 r3 were represented by hyperbolas complying with:
r2 = s2-t2 This hyperbolas that represents different distances from the event horizon (the colored lines in figure 3a) are Lorentz invariant, meaning that an object traveling trough a trajectory along the hyperbola will not fill any difference independently to which point in the hyperbola he is on a determinate time, exactly the same that traveling along the corresponding circumference around the blackhole at a constant radius (see correspondent circumferences at figure 3b) Those hyperbolas conform with:
s = r cosh ω ;
t = r sinh ω
analog to the circular functions x = r cos θ ; y = r sin θ (see figure 4)
t y
ω
θ
s
r
x
r
4a 4b Figure 4 : Accelerated reference frame hyperbolic function compared with normal circle functions Therefore respectively
ds = dr cosh ω + r sinh ω dω ; dt = dr sinh ω + r cosh ω dω Each ω represents a line of equivalent time, while the smaller r the higher the acceleration each observer senses while moving along the hyperbolic trajectory. Page 4/7
The acceleration each observer feels will be:
Where r is the proper distance to the origin Now let us plot 2 observers a and b, and lets call them Alice and Bob (to be original). Alice travels into the blackhole while Bob stay out of it at a constant distant from the event horizon. Bob is concerned about Alice then he observe her at constant clock ticks times. What do he see? t ω=2
s r
Figure 5 : Alice (red) travels to the blackhole while Bob (green) stay away at a constant radius from the event horizon. At each time tick (narrow blue lines) he look to Alice and see her slowing down, see how Bob's constant time increments correspond to smaller and smaller time increments for Alice. Does Alice feel he is slowing down? No, on Alice time frame he is not slowing down This happens because Alice and Bob are at two different time reference frames. Those reference frames are related accordingly with Einstein General theory of relativity by the relationship:
Where: Ʈ : proper time (Alice's time) t : normal time (Bob's time) r : distance to the origin Page 5/7
The solution of this equation shows 2 singularities : 1st when 2MG = r meaning at the event horizon since 2MG = Rsw nd 2 when r = 0 meaning at the singularity itself Lets concentrate on the first singularity: What happens near the event horizon? As r gets closer and closer to Rsw Alice proper time gets slower and slower, finally standing still at (if) she gets to the event horizon. This is a result of the space-time deformation caused by the mass of the blackhole. Let me explain that deformation in the next paragraph Gravitational space-time deformation: As Einstein predicted space-time is deformed by the gravitational field generated by any mass. The curvature of space is in fact on its 3 dimensions (plus time), meaning there is at least an other 4th dimension of space, however since it is hard for our minds to depict 4 space dimensions and even harder to draw it, is common to make the analogical comparison of a two dimensional surface deformation: i.e. to put a ball over a trampoline. (see figure 6) The heavier the object, the deeper the deformation. The dense the object, the higher the deformation curvature will be.
Figure 6 : 2D analogical representation of the gravitational space-time deformation caused by a stellar mass On the case of blackholes, the curvature is so high that is like the “trampoline” is deformed towards an infinite deep. Therefore in order to depict the 1st singularity – the event horizon (see previous paragraph) lets draw the following image (see figure 7) :
Page 6/7
Event horizon
Event horizon
a
a' Figure 7 : 2D analogical representation of the gravitational space-time deformation caused by a blackhole, On figure 7 Alice trajectory is like a “water falling trough a sink hole ant”. A small ant swimming towards the center of the sink hole on the water surface, will never reach its center, at maximum will continue falling parallel to the pipes wall. Analogically on 3D, Alice traveling towards the event horizon (on figure 7 from a to a') has a lower and lower velocity component perpendicular to the event horizon, therefore an observer located outside the blackhole will see her slowing down, while she see constant velocity. So when will Alice cross the event horizon? As time pass and Alice approach the event horizon Bob see her with a larger and larger red shift. Near the event horizon Bob will see Alice almost frozen, its “light” being received at very low frequencies. In fact as the eons pass Bob still don't see Alice cross the event horizon, however time pass.. New stars forms, old starts die, the distance between the galaxies gets bigger and bigger till all the galaxies pass trough the horizon, never to be seen again, and his own galaxy start to dim... till all the stars burn all of their fuel, nothing is left, and she still didn’t cross the event horizon. And what about his blackhole? As all blackholes he also slowly but inevitably evaporates at a faster and faster rate (Hawkins radiation) finally this blackhole also disappears in a blast, when its mass is too small to remain a blackhole, so there is no more event horizon to cross. From Bob's point of view, that means to all the universe outside the blackhole, Alice never reached the blackhole event horizon, less of that never crossed it. Therefore nothing can cross a blackhole event horizon, not in all time of the universe. (and no blackhole will eat us) Thanks: I want to tank Stanford, Yale and MIT Universities for publishing free on the Internet curses allowing people like me to be updated and learn, even when no time nor money will allow other ways, and specially I wish to tank Leonard Susskind for its clear yet advanced and detailed lectures on physics.
Page 7/7
