# Zero Time, Infinite Space?

Gold Member
I am trying to clarify some implications that appear to fall out of the Schwarzschild metric. The inference is that with respect to a distant observer, the tick of the clock slows on approaching a black hole event horizon at [Rs] and space expands in the radial direction. In flat spacetime, we are familiar with the ratio of the radius of a circle to its circumference, which in some standard texts is referred to as coordinate-radius:

$$Coordinate-r = circumference/2\pi$$

Under relative motion, i.e. special relativity, the observed spatial length is said to contract in the direction of motion. However, a comparable change in the observed rate of time is understood to keep the speed of light [c] invariant in both the moving and stationary frames of reference. Time also dilates under gravity, but now we appear to have to resolve the implications of an expansion of space in the radial direction towards [Rs] rather than a contraction. The relativistic factor by which time dilates and space expands under gravity is defined by:

$$\gamma = \frac {1}{\sqrt{1-Rs/r}}$$

Therefore, this would suggest that the true radius or Spatial-radius would be defined as follows:

$$Spatial-r = \gamma(coordinate-r)$$

So the basis of my questions relate to whether this spatial expansion of radius is real and what implication follow from it. Now it is assumed that time dilation is real in the sense that if 2 twins (A & B) are initially located at a distance from [Rs], but twin (B) then approaches the event horizon and returns, (B) would be physically younger than (A). Now on this journey, the implication is also that coordinate-r would become increasingly smaller than spatial-r, as the curvature of space increases under the effect of gravity.

So does light have to travel an ever-longer physical distance between the twins as (B) approaches [Rs]?

If we assume that twin (B) approaches [Rs] and then stops, then only the effects of gravity need to be considered. We might also assume that both twins are firing a laser pulse every second, which is then reflected back off the other craft:

Can the round trip distance be inferred from the speed of light [c]?
What distance [d] would each twin inferred from [d=ct/2]?

At one level, irrespective of the expansion, it would seem reasonable to assumed that the distance between A-B is the same as B-A, at least to the photons, but the implication of time dilation is that the rate of time in (A) and (B) is running at different rates. So

Can we assume the value of [c] is constant with respect to spatial-radius?

Continuing this process all the way to the black hole event horizon, the relative time in (B) with respect (A) seems go to zero and the spatial-r separation would become infinite.

So is it valid to ask exactly when and where does a black hole exists in spacetime?

Footnote:
In part, this thread was raised after coming across the formal definition of a black hole as “a region of spacetime that is not in the causal past of the infinite future.” Not sure that I really understand what this implies either!

At one level, irrespective of the expansion, it would seem reasonable to assumed that the distance between A-B is the same as B-A, at least to the photons, but the implication of time dilation is that the rate of time in (A) and (B) is running at different rates.
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I am not an expert in this but I believe that although it seems reasonable classicaly this is not true in accordance with relativity. As I understand it being in a gravity well is akin to accelorating away. And as the muon expiroment shows (albeit with just velocity not acceloration) there will be length contraction as well. Making a-b not equal to b-a.

Footnote:
In part, this thread was raised after coming across the formal definition of a black hole as “a region of spacetime that is not in the causal past of the infinite future.” Not sure that I really understand what this implies either!
It means that everything inside that region cannot influence anything else in the universe forever.

Gold Member
Response to #2:

I’m not an expert either, so appreciate the response. You may well be right because at some level things don’t seem to add up.

I am not an expert in this but I believe that although it seems reasonable classically this is not true in accordance with relativity. As I understand it being in a gravity well is akin to accelerating away. And as the muon experiment shows (albeit with just velocity not acceleration) there will be length contraction as well. Making a-b not equal to b-a.

If your suggestion is true, does the different gravitational acceleration at (A) and (B) transpose into a relative velocity between (A) and (B)?

If so, on the basis that (B) is subject to a higher gravitational acceleration, does (B) have a higher relative velocity with respect to (A) that causes the additional space contraction that makes A-B and B-A different?

This seems to make things very complicated and would it not also imply an additional time dilation factor?

Gold Member
Response to #3

It means that everything inside that region cannot influence anything else in the universe forever.

Thanks, that makes sense, but I thought black holes evaporated over incredible periods of time suggesting that something inside the horizon does appear back in the main universe?

Thanks, that makes sense, but I thought black holes evaporated over incredible periods of time suggesting that something inside the horizon does appear back in the main universe?

that's true, the black holes do evaporate over immensly huge periods of time. the matter that the balck hole sucks in is eventually leaked out, even though it's not in the form of the atoms, particles, e.t.c that it came in in.

that's true, the black holes do evaporate over immensly huge periods of time. the matter that the balck hole sucks in is eventually leaked out, even though it's not in the form of the atoms, particles, e.t.c that it came in in.

Which is one reason Quantum Theory disagrees with General Relativity (At least the prediction of the GR that black holes have all their mass located in a singularity of infinite density at the centre) because QT says information can not be lost.

Which is one reason Quantum Theory disagrees with General Relativity (At least the prediction of the GR that black holes have all their mass located in a singularity of infinite density at the centre) because QT says information can not be lost.

i've been thinking, if the first dimension is a line, is singularity like a dot? =)

i've been thinking, if the first dimension is a line, is singularity like a dot? =)

I'm guessing that the uncertainty principl excludes having a stationary singularity because knowing its exact velocity means it can not have an exact location, which rules out having a singularity of zero spatial dimensions.

Gold Member
Response to #6:
That’s true; the black holes do evaporate over immensely huge periods of time. The matter that the black hole sucks in is eventually leaked out, even though it's not in the form of the atoms, particles, etc that it came in.

If so, is the formal definition wrong?
The formal definition of a black hole as “a region of spacetime that is not in the causal past of the infinite future.”

Response to #7:

Which is one reason Quantum Theory disagrees with General Relativity (At least the prediction of the GR that black holes have all their mass located in a singularity of infinite density at the centre) because QT says information cannot be lost.

I believe black holes raise some serious questions about the true underlying nature of matter. The concept of zero spatial dimensions and infinite density do not make sense to me, therefore I believe there is still some scope for speculation on these matters. Of course, it should be highlighted that this statement is, in itself, just speculation.

Response to #8/9:

I’ve been thinking, if the first dimension is a line, is singularity like a dot?

I think quantum theory would question everything below the Planck length, 1E-35metre or shorter than the Planck time 1E-43second. Equally, the notion of all our current concepts about atomic structure of 3 millions Sun’s being compressed into these dimensions is still questionable within the context of a particle model.