DrGreg said:
Warning: I am only a beginner in GR, so I stand to be corrected by an expert.
So am I, so expert opinions are welcome
DrGreg said:
a_{proper} = \gamma^3 a_{coord}
is, I think, true for a local inertial coord system (in which SR is approximately true), but I believe it's not generally true in other coords, due to gravitational time dilation and length contraction.
That equation comes directly from the Schwarzschild solution so it relates the measurements of a observer at infinity to that of a local observer. The Schwarzschild solution takes gravitational time dilation and length contraction into account. In fact it defines it.
kev said:
The significance of that equation is that the coordinate acceleration when the radius is less than the Schwarzschild radius is negative and accelerates particles outwards from the centre of a vacuum solution black hole towards the event horizon. The proper acceleration below the event horizon is imaginary while the coordinate acceleration is real indicating the coordinate acceleration is the physical solution while the traditional picture of a black hole as central singularity of infinite mass is not. The coordinate solution shows there is in fact infinite acceleration outwards at the centre of a black hole indicating that it impossible for a true singularity to form. This agrees with another thread where we showed that the interior Schwarzschild solution shows that a region of negative time forms inside a gravitational body before it has collapsed to the density required to form a black hole and that this would be likely to prevent a central singularity forming. This shows that a black hole is in fact a hollow shell that is marginally larger than the Schwarzschild radius with no mass inside the event horizon. The radius of the shell is asymptotically collapsing toward the Schwarzschild radius but never quite reaches it.
DrGreg said:
To my inexpert ears, this sounds nonsensical. Is there a GR expert to confirm this?
I do not take offence that you think it is nonsense, because it is not the mainstream conclusion of numerous books on the subject. On the other hand, it is much easier to sell books describing a black hole as singularity that may be the portal to other universes than the description of black hole as a boring hollow shell of matter that remains in this universe.
More seriously, coordinate systems like those of Eddington-Finkelstein and Kruskal seem to prove that a free falling observer reaches the centre of the black hole in finite time. However, these alternative coordinate systems are derived from the vacuum Schwarzschild solution so they assume a priori that a black hole is a singularity of infinite density so they can not be used to prove that a black hole is a stable singularity of infinite density. None of the alternative coordinate systems seem to take the interior Schwarzschild solution into account. They also seem to assume that proper local measurements are more valid than coordinate measurements. Further on in this post, I hope I have shown that is not necessarily true.
DrGreg said:
Are you aware that the t and R coordinates of the Schwarzschild solution represent the time and space according to a distant observer (theoretically at infinity), and do not represent the time and space of a local observer at the event being measured.
Yes and yes, but technically the Schwarzschild solution contains both local and coordinate measurements. The dtau in the equation is local proper time and dt is the time measured by an observer at infinity. In fact we do not need to be limited to a binary choice of just local observers or an observer at infinity. The gravitational time dilation derived from the Schwarzschild solution can be expressed as:
dt ' = dt \frac{\sqrt{1-R_s/R_o}}{\sqrt{1-R_s/R}}
where Rs is the Schwarzschild radius, Ro is the location of the observer and R is radius where the time dilation is being calculated for. It is easy to see that when Ro=infinity that dt ' = dt/\sqrt{1-R_s/R} and when Ro=R that dt'=dt. The beauty of this formulation is that the observer can be above or below where the measurement is being made so for example when Ro=4Rs and R=5Rs the observer sees the clock above him running faster (blue shift) and when Ro=5Rs and R=4Rs the observer sees the clock below him running slower (red shift).
Similarly length contraction can be expresses as:
dr ' = dr \frac{\sqrt{1-R_s/R}}{\sqrt{1-R_s/R_o}}
dr '/dt ' = dr/dt \frac{(1-R_s/R)}{(1-R_s/R_o)}
d^2 r'/dt^2' = d^2r/dt^2 \frac{(1-R_s/R)^{(1.5)}}{(1-R_s/R_o)^{(1.5)}}
The above expressions for velocity and acceleration do not tell us what the local velocity or acceleration actually is unless we know the coordinate value or vice versa.
We know that the local speed of light is always c so we can say with confidence that:
c ' = c\frac{(1-R_s/R)}{(1-R_s/R_o)}
and from considerations of escape velocity the velocity of a particle with mass falling from infinity is
v' = c\sqrt{Rs/R} \frac{(1-R_s/R)}{(1-R_s/R_o)}
For acceleration the local gravitational acceleration (from sources quoted earlier in this thread) is:
a=\frac{GM}{R^2}\frac{1}{\sqrt{1-R_s/R}} so the trasformation is:
a'=\frac{GM}{R^2} \frac{(1-R_s/R)}{(1-R_s/Ro)^{(1.5)}}
Below the event Schwarzschild radius the above expression for gravitational acceleration inverts and anything that happens to be below the horizon is accelerated outwards according to an observer at infinity. To the local observer it appears as if the acceleration is towards the centre of the black hole. Most of the literature gives priority to the measurements of the local observer (the proper measurements) but I challenge that. While in Special Relativity the measurements made by one observer of a system moving relative to him are ambiguous the measurements made at a distance in General Relativity are not. For example, an observer low down in a gravitational weel see the clock odf an observer above him as running fast while the upper observer see the clock of the observer below as running slower. No contradiction or ambiguity of measurements there. If the lower observer moves up to the higher observer he finds that his clock has indeed recorded less elapsed time than the higher observer. The measurements made at a distance in General Relativity are accurate and unambiguous. Now the lower observer measure the rate of his own clock as running at one second per second and so does the upper observer. Obviously local proper measurements are not very useful. Indeed when the lower observer realizes that his clock is running slower than that of the upper observer (which he can easily prove to himself by meeting with upper observer and comparing elapsed times) he soon realizes his measurement of the local speed of light lower down in the gravity well as c, is in fact an illusion. The above arguments should make clear that coordinate measurements made by comparing the measurements of various observers are the "big picture" and represent physical reality in so far as there is any such thing ;)
A surprising useful conclusion falls out of the Schwarzschild coordinate solution. Run the universe backwards towards the big bang and you eventually get to a time when the scale factor of the universe is such that the mass contained within the universe is contained within the Schwarzschild radius and the universe is effectively a black hole. I think I worked this out to be about a radius of 300 times the size of the Milky Way galaxy.Some people would object to this view and say the universe was never really bigger or smaller but just differently scaled. On the other hand I think everyone would have to agree that the universe must have had a greater density in earlier epochs and that at one time the density exceeded that required for a black hole. Now if we run the universe forward again we have a potential problem because the classical view is that nothing can escape a black hole which implies we must now still be in a black hole. Observations do not seem to support that conclusion. One solution to this problem would be for the universe to gradually seep out the primordial black hole in the form of Hawking radiation but that would mean the universe started as a big wimper rather than a big bang. A simpler solution is provided directly by General Relativity and the Schwarzschild solution. The equation I gave above for the gravitational acceleration shows that if all the mass of the universe was contained within say a radius of one Planck length then there would be extreme acceleration outwards which would be not unlike the rapid inflation that was supposed to occur early on.
DrGreg said:
Thus coordinate acceleration d2R/dt2 does not represent the acceleration measured by a "stationary" local observer. And are you aware that inside the event horizon, t is proportional to local distance and R is proportional to local time?
As mentioned above, I do not believe the "proper" measurements made by a local observer are the best represention of reality. It is also worth noting that the proper acceleration measured by the local observer below the event horizon is the sqare root of a negative number and therefore imaginary, while the same measurement by the observer at infinity is real.
This paper here http://prola.aps.org/abstract/PRD/v25/i12/p3191_1 seems to support my argument as far as repulsive gravity is concerned.
"
To the distant observer, who uses measuring instruments not affected by gravity, gravitational repulsion can occur anywhere in the Schwarzschild field. It depends on the relationship between the transverse and radial Schwarzschild velocities. On the other hand, local observers, whose measuring instruments are affected by gravity, cannot detect a positive value for the acceleration of gravity."
I am aware of the view that time and distance coordinates swap over below the Schwarzschild radius but I am also aware that view was bitterly disputed in a thread in this forum. I am not sure who won ;) Personally I don't think you can shift coordinates around like that, as you would have to define another coordinate system to shift (or rotate) the original coordinate system relative to.
As I mentioned in another thread ,once it is accepted that a black hole is a hollow shell of mass just outside the Schwarzschild radius, the Information Loss Paradox, location of entropy and source of thermal radiation problems all automatically go away and do not require exotic explanations involving holograms, virtual particles and multiple universes.
but there are some important issues/principles at stake here. I can not find anything on the internet that specifically addresses what the local gravitational acceleration would be, which is why I was hoping you could point to a reference on the subject. :shy:
