# Our light cone was a black hole

My calculations show that until about 3.8 Gy ago (z ~ 1.7) the volume of space defined by our light cone had a high enough density to constitute a black hole, in the sense that 2GM > c2r. The further back in time, the more our light cone exceeded that threshold, because the density increases and the proper radius of our light cone decreases as we go back in time before that point. After that time, the volume of space defined by our light cone has not been dense enough to constitute a BH because both the density and the radius of our light cone have decreased, the latter as an inevitable result of the light travel time becoming increasingly constrained.

I'm not sure whether to attach any particular significance to this fact. In theory no light should have been able to escape (or even move one iota outward) from its emission point at our light cone. Since our light cone is calculated to have expanded for the first 6 Gy or so after the big bang, this seems to be inconsistent with being a bona fide BH. For example, in proper distance coordinates light emitted from distant sources (such as the CBR) is supposed to have moved away from us during that initial time period, and begun moving toward us only after our light cone crossed the Hubble sphere at around z ~ 1.6.

Setting aside the theoretical imponderables, I find this situation to be annoying because I want to calculate the instantaneous gravitational time dilation at our light cone at various historical values of z. But the Schwarzschild metric (both the interior and exterior versions) "blows up" when 2GM > c2r. (Square root of a negative number.) Is there any reasonable alternative means for calculating a "pure" instantaneous gravitational time dilation (i.e., no Doppler effects mixed in) under these circumstances?

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George Jones
Staff Emeritus
Gold Member
Setting aside the theoretical imponderables, I find this situation to be annoying because I want to calculate the instantaneous gravitational time dilation at our light cone at various historical values of z. But the Schwarzschild metric (both the interior and exterior versions) "blow up" when 2GM > c2r. (Square root of a negative number.) Is there any reasonable alternative means for calculating a "pure" instantaneous gravitational time dilation (i.e., no Doppler effects mixed in) under these circumstances?

I'm not sure what you're trying to do because I don't know what "instantaneous gravitational time dilation" means, but you can't use the Schwarzschild solution ( mertic and coordinates) in this context. For example, the Schwarzschild solution is a vacuum solution of Einstein's equation, and cosmological solutions are non-vacuum solutions.

Questions about time are answered using the metric, but the contextually correct metric has to be used.

George, I partially agree with your point.

The Schwarzschild metric certainly cannot be used to calculate timelike or null geodesic events in a Friedmann model, because it doesn't take account of the ever-changing cosmic density.

However, if one slices a spacelike foliation of the volume of the universe out to a certain proper radius from us, (like freezing the instantaneous motion of the cosmic fluid with a "snapshot",) it seems appropriate to treat that volume as the interior of a very diffuse massive body with its center at us and its surface out at the radius. Birkhoff's Theorem says we can ignore the gravitational effect of any shells of mass outside that surface. So why can't one apply the interior Schwarzschild metric to that foliation in order to calculate the "static" gravitational time dilation of the massive body's center compared to its surface, at any given instant in time?

Surely the background momentum of the cosmic fluid itself does not cause the instantaneous influence of gravity within this volume to be different than would be the case if the cosmic fluid were actually temporarily at rest, e.g. static.

By the way, it occurs to me that I may be able to avoid having the interior Schwarzschild metric "blow up" by taking many snapshots of the changing cosmic density in small increments of time and then integrating their respective incremental time dilations.

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atyy
I think Birkhoff only applies to vacuum equations.

I think Birkhoff only applies to vacuum equations.
I don't think so, it's frequently used in cosmological contexts. Wikipedia refers to it as a vacuum solution but also says:

"Another interesting consequence of Birkhoff's theorem is that for a spherically symmetric thin shell, the interior solution must be given by the Minkowski metric; in other words, the gravitational field must vanish inside a spherically symmetric shell. This agrees with what happens in Newtonian gravitation."

Presumably this would apply to succession of concentric spheres around our interior Schwarzschild mass.

Edit: One could ask the question the other way round: If a spherically symmetrical "thick" hollow sphere affects the time dilation at inside the shell, how does one compute the effect it has? How about if the hollow shell has effectively infinite thickness?

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Well, Carroll also has the vacuum qualification: http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html. And the inside of a spherical shell is also vacuum.
OK, I accept that the vacuum limitation on Birkhoff may be valid. Which brings me back to the question: so what is the effect on gravitational time dilation, at the center as compared to the surface of a massive body, of a "thick" massive shell surrounding that surface, where "thick" can mean potentially up to infinitely thick?

atyy
OK, I accept that the vacuum limitation on Birkhoff may be valid. Which brings me back to the question: so what is the effect on gravitational time dilation, at the center as compared to the surface of a massive body, of a "thick" massive shell surrounding that surface, where "thick" can mean potentially up to infinitely thick?

Questions about time are answered using the metric, but the contextually correct metric has to be used.

Does anyone know what an appropriate metric might be?

George Jones
Staff Emeritus
Gold Member
Does anyone know what an appropriate metric might be?

A Friedman-Robertson-Walker metric, possibly projected onto appropriate hypersurfaces, and possible in appropriate coordinates.
nutgeb said:
Which brings me back to the question: so what is the effect on gravitational time dilation, at the center as compared to the surface of a massive body, of a "thick" massive shell surrounding that surface, where "thick" can mean potentially up to infinitely thick?

I'm still not sure of what you're trying to calculate. Can this be expressed as the results of particular measurements by particulars observers?

I'm still not sure of what you're trying to calculate. Can this be expressed as the results of particular measurements by particulars observers?
Thanks George, I'll try to explain.

In a homogeneous model universe with $$\Omega = 1$$, I want to calculate the difference in clock rates between (a) an observer at rest at the coordinate origin, and (b) an emitter at distance r from the origin. Observer and emitter each are comoving with their respective local Hubble flow. I want to use the approach, frequently used in the literature, of considering the density of the sphere centered on the origin with radius r and disregarding the effect of all concentric "shells" of mass located outside the sphere. I then want to apply the Schwarzschild solution, either the interior or exterior version as appropriate, to calculate the gravitational blueshift of the light received by the observer, as compared to the wavelength originally measured by the emitter. By the same token I expect the center observer's clock to run faster than the emitter's, which ultimately could be verified by bringing the clocks together and comparing elapsed times.

I thought there would be no dispute that the Newtonian Shell Theorem and Gauss' Law say that mass outside of the shell would have no effect inside the shell and therefore I could calculate the gravitational blueshift without considering that exterior mass. I also thought that Birkhoff's Theorem says the same thing in a relativistic analysis, but I think atyy raises a good question about whether Birkhoff's Theorem really applies since the volume outside the sphere is not a vacuum. (On the other hand, Birkhoff does tell us that the spherically symmetrical recession motion of the mass in the sphere does not itself invalidate a Schwarzschild analysis in the sphere.)

One source I referred to is J. Peacock's http://arxiv.org/abs/0809.4573v1" [Broken] "A diatribe on expanding space". While describing particle motions (p.3) he says:

"the Swiss Cheese universe [provides] an exact model in which the mass within (non-overlapping) spherical cavities is compressed to a black hole. Within the cavity, the metric
is exactly Schwarzschild, and the behaviour of the rest of the universe is irrelevant."

Later when describing the cosmological redshift as a combination of Doppler and gravitational shifts (p.4) he says:

"Combining Doppler and gravitational shifts, we then write

$$1 + z = \sqrt{\frac{1+v/c}{1-v/c}} \left( 1 + \frac{ \Delta \phi }{c^{2}} \right) ,$$

where $$\Delta \phi$$ is the difference in gravitational potential between the point of emission and reception of a photon. If we think of the observer as lying at the centre of a sphere of radius r, with the emitting galaxy on the edge, then the sense of the gravitational shift will be a blueshift: the radial acceleration at radius r is $$a = GM(<r)/r^{2} = 4 \pi Gpr / 3$$, so the potential is thus $$\Delta\phi = -4\pi Gpr^{2} / 6 = - \Omega_{m} H_{0}^{2}r^{2} / 4$$, considering nonrelativistic matter only for simplicity. The gravitational term is thus quadratic in r and has to be considered when going beyond first-order terms in the Doppler shift. To second order, it is exactly correct to think of the cosmological redshift as a combination of doppler and gravitational redshifts (see Bondi 1947 and problem 3.4 of 'Cosmological Physics')."

So maybe the swiss cheese model requires use of a Lemaitre-Tolman-Bondi (L-T-B) metric instead of FRW for the space external to the cavity? Is the gravitational blueshift formula different in the L-T-B metric? Whether the FRW or L-T-B metric is used external to the cavity doesn't seem like it prevents use of Schwartzschild inside the cavity, as Peacock says.

George, my ultimate goal here is to calculate, by the most accurate means, whether the gravitational time dilation exactly offsets the SR time dilation in my scenario when the emitter is moving radially outward at the Newtonian escape velocity of the mass inside the sphere. If I could apply the external Schwarzschild metric without modification, the two effects do indeed exactly cancel when the emitter is moving radially outward at exactly the escape velocity. But for the reasons discussed earlier in this thread, the internal Schwarzschild metric may be more appropriate. The interior metric is too complex to compare directly to the SR time dilation equation. So I thought I'd just run test numbers in a spreadsheet and see what they tell me. Before I do that I'm hoping you can tell me whether what I'm doing is mathematically valid.

As an aside, when I compared our past light cone to a black hole earlier in this thread, I realize that the regular Schwarzschild metric has no mechanism to cope with the possibility that the BH begins with all (or at least many) of its particles having a strong outward momentum in excess of the speed of light, which is what the FRW metric seems to handle intrinsically. The situation is somewhat analagous to a spinning BH where the centrifugal pseudo-force can theoretically prevent a BH from forming at all, above a certain angular velocity / mass. Of course a spinning BH introduces rotational frame-dragging which would be inappropriate in my scenario.

I don't think I can use the FRW metric for my analysis because FRW seems to rule out the possibility of calculating any net relativistic time dilation, because all fundamental comoving observers measure exactly the same proper "cosmological time". That is what makes me think that the right answer is for gravitational time dilation and SR time dilation to exactly cancel each other at escape velocity. A spatially flat FRW metric is characterized by recession rates equal to escape velocity.

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Ich
Hey, great, this paper by Peacock expresses quite exactly my own thoughts on the subject. You can use it as a more reliable source than me for my claims https://www.physicsforums.com/showthread.php?t=301459"in #79 and #83.

p.s.: In empty space, where Phi =0, you're left with standard time dilatation in Peacock's formula.

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Hey, great, this paper by Peacock expresses quite exactly my own thoughts on the subject. You can use it as a more reliable source than me for my claims https://www.physicsforums.com/showthread.php?t=301459"in #79 and #83.
I agree that you are correct on this point. For 2 particles launched from the same location with a time interval between, the proper distance between them will not increase as a function of time. In fact, in an $$\Omega=1$$ universe the proper distance between the 2 particles will decrease, as the gravity of their respective background dust "spheres" causes them to accelerate back toward the coordinate origin, with the particle launched first having the larger sphere at each instant and therefore the greater backward acceleration. (They will continue to have residual proper momentum away from the origin for at least as long as the universe continues expanding.)

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OK, while I'm waiting for George to hopefully jump in, I'll try to answer a couple of my questions.

First, I am pretty confident that Birkhoff's theorem can be used (like the Newtonian shell theorem and Gauss' Law) to prove that in an FLRW universe, an arbitrary sphere defined at radius r around a coordinate origin will not be affected by the gravitation of the (potentially infinite) amount of dust outside the sphere. The logic goes like this:

Excavate a cavity with radius r from the FLRW universe. Compress the mass from the cavity into a point mass at the center. Birkhoff's theorem confirms that the metric of the cavity interior is exactly the Schwarzschild (exterior) metric. The central point mass is surrounded by a vacuum, as required by Schwarzschild, up to the radius r we have defined.

Einstein and Straus developed the Swiss-cheese model in 1945 which makes just this assumption. Einstein showed that the metric in the interior of the cavity is exactly Schwarzschild while the exterior metric remains FLRW. His purpose in this exercise was to describe a theoretical basis for why the expansion of the universe does not affect local orbits within our solar system or within our galaxy (causing them to become unstable). When the interior metric is exactly Schwarzschild, by definition the metric in the cavity is governed entirely by the point mass, and mass exterior to the cavity has no effect.

I have read that the L-T-B metric can be used instead of the Schwarzschild metric inside the cavity, if one wishes to model a dynamically increasing radius rather than a static one. For my limited purpose, I will content myself with the static radius of the Schwarzschild metric, taken in static snapshots at intervals over the FLRW expansion.