Flat spacetime in a gravity well

[DaveC426913]

He should start with Newtonian gravity, and the difference between gravitational force and potential, as shown here:

Ah! Now we're getting somewhere.
How does gravitational potential manifest in Newtonian gravity?
What properties does it have that can be measured?

(So far, the only effect of GP I have been able to find is Einsteinian time dilation, thus my quandary.)

 

[Jonathan Scott]

In Newtonian gravity, the difference in potential (typically ##\phi = -Gm/r##) determines the difference in potential energy of an amount of mass in the relevant location compared with the reference location. In General Relativity, that is replaced with a fractional difference in energy due to time dilation, and to make it dimensionless it can be expressed in terms of change in energy for a given rest energy (typically ##\phi/c^2 = -Gm/rc^2##). So the relative fractional time dilation is the difference in the Newtonian potential in dimensionless units.

(Unless you are near a neutron star or worse, the potential is so tiny that the fractional changes in time rate effectively add linearly, as for Newtonian potentials).

 

[DaveC426913]

In Newtonian gravity, the difference in potential (typically ##\phi = -Gm/r##) determines the difference in potential energy of an amount of mass in the relevant location compared with the reference location.

So, a body falling toward it would gain kinetic energy as it fell. But that happens anyway, as it approaches the mass (along the curve).
What measureable effect would it have once in the hollow sphere?

Oh. I think I see.
In my colleagues's graph:

the implication of the cusp and vertical slope is that any mass would instantly get stuck to the inner surface of the sphere - where it has minimum potential energy. It would take, essentially escape velocity (i.e. enough velocity to get back to flat space at an infinite distance) to break away from the inner wall of the shell and head toward the centre.

 

[PeterDonis]

What about hovering at the midpoint between two identical, non-rotating black holes? My intuition at least tells me (if the black holes are close enough, even momentarily) that there will be significant time dilation, even though the net “gravitational force” will be 0

Your intuition is correct as far as it goes. However, there is a key difference between this situation and the situation of a single gravity well: this situation is not static. The holes will fall towards each other. That means that, strictly speaking, the concept of "gravitational time dilation" is not well-defined in this scenario.

It is possible to define "gravitational time dilation" for the specific case of the observer "hovering" at the midpoint between the holes--at least for the period before that observer crosses the horizon of the combined hole formed by the merger of the two holes falling together. This is because the observer "hovering" at the midpoint, while he remains outside the horizon of the combined hole, is at rest relative to an observer at infinity. But this definition does not generalize to other observers.

 

[sweet springs]

Your friend at the center of the Earth lives a daily life with normal time speed. He can cook boiled eggs in ten minutes of his watch and he will live 80 toghether with his old pendulum clock. There is no time dilation in his life.

Time dilation works when we compare his time and our Earth surface time. How can we compare?
We can use light signal. He will emit light of 1.0Hz to us. We receive it as 0.6Hz for example. because energy of light decreases traveling in gravity. Blue light turns to be red light, for example.

The broadcast he sings we see and find he sings very slow with ratio of 0.6/1.0. We find his time is delaying.
We send a broadcast of singing to him. Radio waves got energy by going down the well. It becomes shortened and condensed. He will listen a singer sings very quick. He find that our time goes quickly.

So in short, though no gravity force applies on him, emitted light has to go up the gravity well and lose energy and thus frequency, that we observe his pace of time is slow than ours.

 

[Eric Bretschneider]

The membrane version of gravity wells leads everyone to assume that the slope is directly associated with the strength of gavity, hence at the "bottom" where the slope is you have no gravitational forces and flat spacetime.

In terms of relativity its more of a stretching effect. The larger the distortion the greater the impact on relativistic parameters (time dilation). The presence of mass distorts spacetime (stretches it). It is stretched/distorted the most at the center of the mass. It might be worth trying to view a uniform grid through a lens, but not with everything at the focal point. Put a half spherical lens on your grid and the distortion can approximate what happens to spacetime. It will be distorted the most over the center of the lens.