UPDATE October 29, 2021: We updated the text to reflect that the theorem of our October 16, 2021 blog post is incorrect.
----
Our view is a new interpretation of general relativity. We expect our interpretation to be completely equivalent to the standard geometric interpretation of general relativity.
If our interpretation follows the mathematics of the Einstein-Hilbert action, then it will certainly reproduce the same effects as the standard interpretation. However, we have not proved that our interpretation totally conforms to the Einstein-Hilbert action.
Let us look at some questions which we have raised in the past two weeks.
Lowering down of a mechanical clock in a Schwarzschild potential
We claim that the standard Schwarzschild coordinates r and t are the "true" coordinates of the global flat Minkowski metric outside the event horizon. For an observer in the potential well, proper time has slowed down, and proper radial distances are stretched, relative to the global Minkowski coordinates.
The apparent slowing down of time deep in the gravity potential well is due to:
1. the increased inertia of a mechanical clock part and
2. the weakening of forces (the gravity between the central mass and the clock does not weaken, though).
The movement of a clock part moves large energy in the gravity field, which causes extra inertia. The extra inertia is many times the original inertia when we are close to the horizon.
The energy which is used to start the clock has done work when we lowered it down, because the energy was pulled by the gravity field. Consequently, there is less energy available to move a clock part.
If the potential is -1% * m c², then the horizontal inertia is 1% larger and forces are 1% weaker. The clock ticks 1% slower.
When a photon dives down in the 1% potential, it retains its energy since the work it did goes to its own (kinetic) energy. The inertia of the photon grows by 1% in horizontal movement, and the horizontal speed of light is 1% lower.
In vertical movement, the inertia is 2% higher, and the measuring rod has contracted 1%. A clock, whose parts do vertical movement, ticks as fast as a clock whose parts do horizontal movement.
A photon falling down to the event horizon: it typically passes the horizon in less than a millisecond of global Minkowski time
In the standard Schwarzschild coordinates, the proper time relative to the Schwarzschild coordinate time has slowed down by a factor
sqrt(1 - r_s / r),
where r_s is the Schwarzschild radius of a central mass M (the mass is measured from far away), and the observer is at the distance r from the center in Schwarzschild coordinates.
Proper radial distances have stretched by the inverse factor
1 / sqrt(1 - r_s / r).
Therefore the radial speed of light in Schwarzschild coordinates is slowed down by the factor
1 - r_s / r.
The coordinate time t (not proper time) that it takes a descending photon to reach the horizon, if it starts from R, is
R
t = 1 / c * ∫ 1 / (1 - r_s / r) dr
r_s
R
= 1 / c * ∫ r / (r - r_s) dr
r_s
R - r_s
= 1 / c * ∫ r_s / x dx,
0
if R is only slightly larger than r_s. We have denoted x = r - r_s.
The elapsed coordinate time t diverges logarithmically. As if the photon would float forever close to the horizon.
However, in the Minkowski & newtonian model the slowdown of light is due to the extra inertial mass it gains from the field of the black hole, and that inertial mass cannot be infinite.
The smallest possible factor for the slowdown of light is
m / M,
where m is the energy of the photon divided by c². The largest possible value of the integrand above is M / m. That is reached when x = r_s m / M. We calculate the corrected integral separately for [0, x] and [x, R - r_s].
The corrected value for the integral is
t = 1 / c * M / m * (r_s * m / M)
+ 1 / c * r_s * ( log(R - r_s) - log(r_s m / M) )
= r_s / c
+ r_s / c * ( log(R / r_s - 1) - log(m / M) ).
Since -log(m / M) is typically a large number, 100 or more, we can ignore smaller terms and write:
t = r_s / c * log(M / m).
An example: we let a 1 MeV photon fall into a solar mass black hole. The logarithm is then log(10⁶⁰) = 138. It takes at most
t = 1.38 ms
of global Minkowski time for the photon to fall through the horizon. We cannot really say that the photon stays "floating" at the horizon. Floating is an idealization which would happen at an infinite mass black hole, or if the photon has infinitesimal energy.
The collapse of a uniform ball of dust
According to our calculation in the previous section, a large mass will fall through a horizon in a millisecond or less of the global Minkowski time.
Furthermore, our "syrup" model of gravity suggests that in a large collapsing mass, light is dragged along the mass. Even though light slows to a crawl relative to the mass, the mass itself can move fast as seen in external Minkowski coordinates.
We believe that the combination: external Schwarzschild metric and internal FLRW metric describes the collapse of a uniform ball of dust (an Oppenheimer and Snyder collapse) in our Minkowski & newtonian gravity model. That is, the description of general relativity is correct.
Question. What are the "true" global Minkowski coordinates for the interior of the collapsing ball? That is, if we take into account complex effects of the newtonian gravity field.
In the case of the curvature k = 0, proper distances inside the FLRW part probably are the spatial global Minkowski coordinates. But what is the time coordinate?
In our calculation in the previous section we assumed that a single photon can acquire the entire mass of the black hole as its inertial mass. That probably is an exaggeration.
Singularities
It looks like that the Minkowski & newtonian model creates the same singularities as general relativity.
That is not a big surprise because a uniform dust ball does collapse to a singularity in the classic, year 1687 version of newtonian gravity.
If the dust ball has passed its Schwarzschild radius, gravity, apparently, can "fool" other forces. If another force resists contraction, gravity can lower the potential of the collapsing mass, and in that way provide whatever energy the other force is demanding.
Birkhoff's theorem states that a spherically symmetric collapse has to keep contracting if it has passed the Schwarzschild radius. All light cones point downward. The only way that gravity can prevail over a stronger resisting force is that gravity must fool that other force.
Thus, the Minkowski & newtonian model does not save us from singularities. However, uncertainty principles seem to prevent singularities, according to our previous blog post.
The Big Bang is much like a reverse collapse of a uniform ball of dust, in both general relativity and the Minkowski & newtonian model. Quantum mechanics saves us from a singularity there, too - at least in the Minkowski & newtonian model.
No comments:
Post a Comment