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To the question -whether the velocity of light is only constant in a space where the gravitation potential is constant- is constant, John Rennie answers that “The coordinate velocity of light can be different from $c$. The local velocity of light is still $c$.”

According to relativity theory a clock is observed to run at a slower pace as the gravitational field at the clock is stronger (the gravitational potential higher) than it is at the observer -the clock and observer are at rest relative to the source of the field, a neutron star, say. As the speed of light is the same everywhere as measured locally, this means that a measuring rod near the clock inside the gravitational field is observed to be shorter.

I read somewhere “We already know the concept of gravitational potential energy (the metric tensor in GR). For a reference mass/energy this could be interpreted as measure of the "density" of space time, with differentials representing areas of higher curvature, and higher apparent gravitational force. But it would be wrong to suggest [that] the gravitational potential (not the rate of change of gravitational potential) is proportional to the curvature/force.”

My question is: Though the coordinate speed of light then decreases as it enters the gravitational field of a neutron star as observed from outside the field, from a position where the field is weak, can we say as well that it takes light a longer time to travel toward the neutron star because the spacetime distance between the observer to the surface of the neutron star -or, if we could drill a tunnel through the neutron star and keep it open- the distance between the observer and the center of the neutron star is greater as measured locally, inside the gravitational field of the star (I assume that this is the proper distance between the observer and the center of the neutron star?) than as measured outside the field, than the coordinate distance between the observer and the center of the star, their distance as calculated from the positions of the observer and neutron star relative to surrounding stars?

If so, then does this mean that spacetime isn’t so much curved in the sense that spacetime near the neutron star becomes more dense (relative to areas farther from the neutron star, farther from masses, but that, while the mass of the neutron star does curve spacetime in its environment, its gravitational field nearer to the neutron star can be thought of as a local extension of spacetime, an area which unfolds to a photon penetrating its field, traveling toward the center of the neutron star?

Put differently, does mass only deform the texture of spacetime in the sense that if we were to imagine empty space to be furnished with a regular 3-dimensional grid (the cells, the cubes of which are 1 x 1 x 1 meter), would the grid only deform near the neutron star if we were to put the star in empty space -that is, would the number of cells remain unchanged- or would there appear new cells near the neutron star -cells which look smaller to the distant observer than they were in the absence of the neutron star at the same place -that is, at the same coordinate distance?

While I have been asking related questions here and here but they keep getting closed for some reason, in the second case I am referred to this question the answer to which is quite different from what I wanted to know. Though I suspect that the possibility that the neutron star’s gravitational field in its near vicinity is a local extension of spacetime isn’t accepted physics, I still would very much like an answer so hope that this question isn’t closed as soon as I post it as well as my two previous questions.

@safesphere: I don’t mean an expansion of space in time -expansion as a verb- but an extension of spacetime which would be observable by the observer with a powerful telescope if he could observe such grid. My question is whether a spacetime which already exists in the absence of the neutron star just warps, modifies the grid if we insert the neutron star as in this picture so the number of cells remains unchanged, or whether its gravitational field, nearer to its center constitutes a local extension of spacetime: whether the number of cells there is greater after putting the star in it, their size smaller according to the observer than the cells at the same coordinate distance in other directions than that of the neutron star.

If so, if it is an extension of spacetime, then doesn’t this mean that the creation of energy (in this case of the neutron star, of its neutrons or quarks) is (accompanied by, impossible without, indistinguishable from) the creation of spacetime? If so -if space, time and energy aren’t unrelated quantities- then how can we speak about the energy density of the universe (which is supposed to determine its rate of expansion), a concept which seems to define energy and space as independent quantities -which if true would mean that localized energy cannot curve spacetime?

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  • $\begingroup$ Your question is unclear. By the “local extension of spacetime”, do you really mean the radial space expansion? If so, then your logic would be correct. In the Schwarzschild metric outside a non-rotating neutron star, the time dilation is reciprocal of the radial distance expansion. Can you please clarify the question? $\endgroup$
    – safesphere
    Jul 5, 2020 at 17:31

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Instead of a Neutron star, it is easier to think of Rindler observers, because the spacetime is flat and the complications of curved spacetime can be avoided. But the gravitational effects are all there.

A row of accelerated ships keeping the same distance defines a grid, until the horizon, (corresponding to an infinite acceleration), and that has also a defined distance from each of them.

Any object falling from the ship with acceleration $g$ for example will reach the horizont after some time for any inertial observer. But for any of the ships, it never reaches it. It's light signal takes more and more time to go until each of them, as it comes closer to the horizon.

The speed of light is $c$ for any inertial observer and also locally for any of the accelerated ships. But it decreases from each ship to the horizont, as measured by one of them.

The ships and a momentarily comoving inertial observer agree about the distance between them (the grid). But disagrees about the time it takes to reach the horizont, and consequentely about the global speed of light.

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  • $\begingroup$ The OP has asked about a curved spacetime, not the flat Rindler spacetime, so your answer is not applicable. You start with a wrong premise that the Rindler spacetime Is easier to think of compared to the Schwarzschild spacetime. This is obviously false and leads you to a number of inaccuracies. For example, “A row of accelerated ships keeping the same distance” - This implies the same acceleration, so each ship is at the same distance from its horizon. Or: “will reach the horizont after some time for any inertial observer” - There is no horizon for an inertial observer. $\endgroup$
    – safesphere
    Jul 5, 2020 at 17:50
  • $\begingroup$ @safesphere It is easier to deal with 2 dimensions than 4 dimensions and spherical coordinates. If tidal forces are not important (as it seems in this question) uniform acceleration is a good model to understand doubts about spacetime. $\endgroup$ Jul 5, 2020 at 18:44
  • $\begingroup$ The Schwarzschild spacetime is “curved” in 2 dimensions, $r$ (radial distance expansion) and $t$ (time dilation). I agree with your reference to the equivalence principle, but the Rindler coordinates may be more confusing than Schwarzschild for a number of reasons. For example, there is no event horizon in the Rindler spacetime. Instead the horizon is apparent and different observers disagree on its location and existence. $\endgroup$
    – safesphere
    Jul 5, 2020 at 21:22

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