Time Dilation in Gravitational Field: Potential vs Field Strength

[Green dwarf]

I'm wondering is whether it is the gravitational potential (in J/kg) at a point in space that determines the rate of passage of time, or whether it is the gravitational field strength (in m/s2).

To clarify, suppose you had a very heavy hollow spherical shell. The gravitational potential would be lower inside the shell than outside, though the gravitational field strength inside would be zero.

If it is the potential that slows down time, then time would go slower inside the shell than outside; if it is the field strength, time would go faster inside.

I'm wondering about a black hole where it seems to me that matter falling in would stop at the event horizon (from the perspective of someone outside, because time stops there). This would mean that the matter in a black hole forms a spherical hollow shell. I'm probably wrong there. Someone might like to correct me.

 

[Dale]

It is the potential that determines gravitational time dilation.

If it is the potential that slows down time, then time would go slower inside the shell than outside; if it is the field strength, time would go faster inside.

Yes. Time would go slower inside the shell

 

[Green dwarf]

Thanks Dale. I'm wondering too, would there be any truth in the idea that the matter in a black hole would form a thin shell on the event horizon because matter falling in would stop there (from an outsider's perspective)?
One might think that as the black hole grows, matter inside it would then move towards the centre.
But, the gravitational potential should continue to increase (beyond the point where time stops) as you move towards the centre (assuming there is some matter inside). I might then naively think that time might reverse inside the black hole and this might lead to the matter inside moving outwards, back towards the event horizon, thus tending to keep the black hole hollow.
Armchair speculation only, but I would be interested in your comment.

 

[Nugatory]

I'm wondering too, would there be any truth in the idea that the matter in a black hole would form a thin shell on the event horizon because matter falling in would stop there (from an outsider's perspective)?

No.
That’s a common misconception, and we have many threads here that explain why if isn’t that way. But for a quick sound bite sort of answer…. There’s a big difference between “it doesn’t fall through the even horizon” and “it falls through the event horizon but light from this event never reaches our eyes”.

 

[Orodruin]

To expand a bit on that, the gravitational time dilation formula assumes a static observer (the one just containing the gravitational potential is a further simplification valid in the Newtonian limit). An observer cannot be stationary at the event horizon since it is a null surface. The entire idea of ”time stands still at the event horizon” is therefore in the same category of misunderstandings as ”a photon experiences no time”.

Black holes are also extreme objects with significant spacetime curvature. Thinking about space as you normally would around a black hole is a misrepresentation in itself.

 

[Dale]

But, the gravitational potential should continue to increase (beyond the point where time stops)

The gravitational potential is only defined for stationary spacetimes, and inside the event horizon the spacetime is not stationary. So it does not continue to increase, it just stops having meaning altogether.

 

[PeroK]

I'm wondering about a black hole where it seems to me that matter falling in would stop at the event horizon (from the perspective of someone outside, because time stops there).

Time does not stop at the event horizon. It'strue that there are many sources on the Internet that claim this, some of which are seemingly reputable. But, they are all wrong.

If you drop an object into a black hole then send light signals after it, you may calculate:

1) the last light signal that reaches the object before it crosses the horizon.

2) the further light signals that reach the object after it crosses the horizon and before the singularity.

The calculations are here:

https://www.physicsforums.com/threa...alling-into-a-black-hole.1012103/post-6599762

 

[Ibix]

I'm wondering about a black hole where it seems to me that matter falling in would stop at the event horizon (from the perspective of someone outside, because time stops there).

"Time stops at the event horizon" is the same kind of mistake as saying that if you walk due north to the pole then, since there's no more north, you can't carry on walking forwards through the pole. The complexities of geometry in curved spacetime mean that the "time stops" myth is not as blatantly silly as the north pole analogy and it did take us a few years to figure it out, but it's been understood since 1933. It's not clear to me why it gets repeated still.

 

[Orodruin]

It's not clear to me why it gets repeated still.

Same reason relativistic mass gets repeated.

 

[Green dwarf]

Thank you everybody for your responses. As I expected, the situation is much more complicated than the simple model I was imagining. I really don't understand general relativity and probably never will. At 70 years of age, I'm probably better suited to growing beans in my vegie patch.
I do run a school astronomy club for kids aged 10 to 17 though, so I like to find stuff that will fascinate them. My aim is to stir up interest rather than to impart reliable knowledge. If any become really interested, they will get the knowledge themselves later. They do like the song 'What's inside of a black hole?'

 

[Ibix]

Some Johnny Cash influence there, I think.

There's nothing inside a black hole and its internal volume is infinite. The singularity is not a point at the center but is more like a point in time - once you are inside, you can no more avoid it than you can avoid Monday morning. That's all assuming GR is correct, but we strongly suspect it goes wrong somewhere in there, so take with a pinch of salt.

 

[Sagittarius A-Star]

I'm wondering is whether it is the gravitational potential (in J/kg) at a point in space that determines the rate of passage of time, or whether it is the gravitational field strength (in m/s2).

Time dilation depends on the chosen reference frame.

What does not depend on a reference frame are for example different aging in the twin paradox scenario or observed blue-/red-shift of light, that comes from an optical clock.

For example, consider the following local scenario on the surface of the earth:

An observer $A$ stands near the ground level of a building and a lamp is attached to the 5^th floor of the building. The observer $A$ sees a light-pulse from the lamp blue-shifted.

• In the rest frame of observer $A$, the observed blue-shift is due to "gravitational time-dilation" at a location with higher gravitational potential than that of the observer.
• In a free falling reference frame however, the house, the observer $A$ and the lamp are accelerating upwards. The observer $A$ receives the light-pulse while being at a higher velocity into the light, than the velocity of the lamp, when it emitted the light pulse. The observed blue-shift is in this reference frame due to longitudinal Doppler effect instead of "gravitational time-dilation".

 

[vanhees71]

...and it was measured for the first time using the Mössbauer effect by Pound, Rebka and Snider in the early 1960ies.

https://en.wikipedia.org/wiki/Pound–Rebka_experiment