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SpaceProperTime Diagrams and the Central Singularity

Posted August 24, 2020 7:15 AM by Jorrie
Pathfinder Tags: einstein general relativity

In my prior Blog post on the Einstein equivalence principle, I have not identified important values, like the mass of the black hole, real distances and the actual times that would have been recorded by Alice and Bob.

For a BH to have an event horizon radius of 0.5 lyr, it must have an ultra large mass, something like half the mass of the Milky Way. Its Schwarzschild radius must be of order the radius of the Oort Cloud (where the long period comets are thought to come from). It must be similar to this observed BH: The monster black hole in galaxy cluster Abell 85 is roughly the size of our solar system, but packs the mass of 40 billion suns.

Figure 1: Annotated SpaceProperTime Diagram

Alice is parked at a distance of half a lyr from the event horizon, which itself is half a lyr from the singularity, both distances measured as Schwarzschild radial coordinates.[a]

An interesting tidbit: Alice and Bob will feel only 0.35g of acceleration - the more massive a BH is, the lower the g-forces required to hover at a given factor outside the horizon radius.

The original plan was for Bob to free-fall from rest to meet his fate at the singularity. Alice did however convince him that the result will almost certainly be death at the singularity, so they decided to rather drop a well-censored probe into the BH and try to get as much information off the freefall as possible.

Bob equipped both the ship and the probe with a wide-band, variable frequency transceiver and then frequency locked each receiver to the other side's transmitter so that they can track the changes in received carrier frequencies. Bob will ask "Omni-eye" to track the frequency changes and report it to them instantaneously. With that, he gently drops the probe over the side. Let's call this time To.

Immediately Bob notices a redshift in the probe frequency that he receives, which he knows must come from the velocity redshift and a gravitational redshift combined. The probe is both moving away from him and entering regions of lower gravitational potential.

Bob also checks Omni's report on the corresponding frequency that the probe has received. It shows a blueshift, despite the fact that the distance between the ship and probe is continuously increasing. Bob knows that the probe 'sees' their signal blueshifted, but that the velocity redshift diminishes it somewhat, even though the probe's relative speed is still very low. The relative speed between probe and ship increases rather rapidly, but so does the gravitational blueshift. Which one of the latter two will win out?

It so happens that at the event horizon, both red- and blueshift factors are supposedly approaching infinity, so the question is, which one changes faster? We can obviously just wait until Omni-eye tells us the experimental result, but it is fun to try and predict that in advance. The secret lies in the fact that warping of space and time are rather different in this case. Below are two views of the above SpaceProperTime Diagram, rotated so that we look at the curve along selected axes. You can try this with the HSPT.js app that I have posted before.

If we just look outside the BH (blue dots), one can see more severe space warping on the left (into the hyper-plane) than one can see time warping on the right. Now the space-warp determines the blueshift and the time-warp determines redshift that the probe observes. It is clear which one wins.

At the moment that the probe passes the event horizon (yellow dots), space warp is about double the effect of the time warp. The result is that the probe observes Bob's carrier frequency with a 2:1 blueshift ratio. Interestingly, for a quiescent black hole, this will be our probe's only means for detecting the moment of passing the event horizon, at To+1.8 yrs, as annotated on Fig. 1.

For Bob in the spaceship, that moment is not readily detectable due to the double redshift (gravitational and velocity). The probe's signal will be gradually redshifted into non-detectability, without any neatly defined moment of passing the horizon. But fortunately, Omni can to tell Bob that the time is To+2.3 yrs. Omni can also give running commentary on the probe's progress on its way to the singularity.

The probe's frequency will still show a diminishing partial cancelling of the gravitational blueshift by velocity redshift. Eventually the blueshift will overwhelm the redshift and the probe's received frequency will rise exponentially to a death-scream, possibly being fried by the intense infalling energy, while at the same time being pulled apart vertically by the unbounded tidal gravity near the singularity. The last probe report will be at To+2.2 yrs probe time. Bob's ship clock will read 3.4 yrs when he receives the final 'instantaneous' report from Omni.

Bob is now very thankful to Alice for convincing him not to jump in himself...

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Re: SpaceProperTime Diagrams and the Central Singularity

08/24/2020 11:34 PM

" the more massive a BH is, the lower the g-forces required to hover at a given factor outside the horizon radius."

This is certainly not intuitive to me. Can you explain why?

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Re: SpaceProperTime Diagrams and the Central Singularity

08/25/2020 3:16 AM

Sure, it is not intuitive, because one may be thrown by the "a given factor outside the horizon radius", i.e. by a certain percentage outside of the horizon. In this case study, the ship hovers at twice the Schwarzschild horizon radius. The horizon radius is given by Rh = 2GM, so it puts Alice and Bob at r = 4GM.

Using Newtonian gravitational acceleration, it falls off with the inverse square law over distance, i.e. acc = GM/r2, which if we plug in the r = 4GM, gives acc = GM/(4GM)2 = 1/(16GM). This is inversely proportional to mass (for a constant factor outside horizon).

In a nutshell, since the horizon radius grows linearly with the mass and the g-force follows the inverse square law, it drops much faster than the growth in horizon radius, given a fixed % outside of it.

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Re: SpaceProperTime Diagrams and the Central Singularity

08/25/2020 10:15 AM

Thanks Jorrie, that clears that up, but I am also confused by "At the moment that the probe passes the event horizon (yellow dots), space warp is about double the effect of the time warp." I thought it was space-time that was warped, not space or time individually, but maybe you didn't mean that.

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Re: SpaceProperTime Diagrams and the Central Singularity

08/25/2020 10:56 AM

In a gravitational field, it is always both, but there is nothing to stop us from viewing them as two components of the warp of spacetime. When we see the typical "trampoline with the heavy ball at the center" depiction, it is only space-warp that is portrayed. The Einstein equivalence principle (as discussed in prior Blog post https://cr4.globalspec.com/blogentry/29911/SPT-Acceleration-and-the-Equivalence-Principle), depicts only time warping. I found it insightful to take the 'tour' through these steps to arrive at warped spacetime.

There is somewhat more to warped spacetime than just what have been discussed so far in terms of space-propertime, since we have only gone as far as non-rotating, non-charged BH's. I have Blogged on rotating black holes before, e.g. https://cr4.globalspec.com/blogentry/1372/Rotating-Black-Holes-the-Naked-Truth and https://cr4.globalspec.com/blogentry/1670/Extreme-Frame-Dragging, but that was long before I thought of using the explanatory powers of space-propertime.

One problem is that we will need one more spatial dimension to fully exploit that - seeing the rotation in space is not enjoyable with only one space dimension. But there may be ways around it...

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