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I have read these questions:

Does a "Photon Box" have gravitational mass?

Does a box containing photons have more inertia than an empty box?

And it made me curious. i understand that a photon box, though the walls do not have rest mass, and the photons neither, the box itself will still have rest mass, and stress-energy, causing gravitational effects, like bending of spacetime.

But none of these talk about GR time dilation, which is partly caused by the same effects, but this is more an effect of gravitational potential, and the Cristoffel symbols might be different.

The confusion arises, because on this site, there is a question about being weightless at the center of gravity (true), but still, GR time dilation is felt at the center of gravity. The reason for this is that the bending of spacetime is calculated for the center point, where you feel weightless, because effects equal out from all sides. Now GR time dilation is calculated for the area around the center, and therefore is never zero, so yes there is GR time dilation even at the center of gravity.

Now what causes a problem with the photon box, is the center of gravity. The photon box does not have matter in it, just energy, and so what would be the center of gravity? The geometrical center of the box?

Question:

  1. Does the photon box have gravitational potential, and does it cause the effects of GR time dilation?

  2. What will be the center of gravity for the box, where you feel weightless, but GR time dilation still works?

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(1) Yes for both

See, the gravitational time dilation happen because of curved spacetime. It does't matter whether the source of curvature is matter or energy or both.

(2)Geometrical center

The center of gravity is defined with respect to the gravity, irrespective of whether it is caused by energy or mass or both. And is the point where gravitational field converges. So, in this case, because box is massless the only thing which matters is its shape or its geometrical center.

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  • $\begingroup$ Isn't center of gravity only applicable to newtonian approximation? $\endgroup$
    – jinawee
    Commented Oct 1, 2018 at 7:08
  • $\begingroup$ @jinawee Because Newtonian gravity came first, COG is first defined in that context. But it doesn't matter now, because the notion of center of "gravity" can be apply to any system whether the gravity is caused by mass only or not. $\endgroup$
    – Aman pawar
    Commented Oct 1, 2018 at 7:14
  • $\begingroup$ So how would you calculate the center of gravity for an arbitrary metric? $\endgroup$
    – jinawee
    Commented Oct 12, 2018 at 11:27
  • $\begingroup$ @jinawee Mathematically i don't have an answer for a metric but physically you know what it means. $\endgroup$
    – Aman pawar
    Commented Oct 12, 2018 at 14:43

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