According to Einstein, mass, say in the form of matter, curves space. It is the curvature of space that gives rise to gravity. Now I have heard there is an energy associated with the curvature of space. If so, it will have a mass equivalent. Is this mass in addition to the mass causing the curvature (perhaps it is akin to potential energy) or is it the same mass?
$\begingroup$
$\endgroup$
6
-
$\begingroup$ Einstein's Field Equations equated the curvature of spacetime with the energy-momentum stress tensor. i.e. it is energy and momentum that is curving spacetime. Mass is only there because there is a relationship between energy and mass, and that the rest mass energy usually dominates the energy contribution. You should not confuse yourself with the mass concept in GR. $\endgroup$– naturallyInconsistentCommented May 9 at 11:10
-
1$\begingroup$ The short answer is 'no'. $\endgroup$– Martin C.Commented May 9 at 11:18
-
$\begingroup$ But in the FLRW metrics, the mass or energy of curvature appears in addition to the energy or mass of all the other components. $\endgroup$– John HobsonCommented May 9 at 11:22
-
$\begingroup$ What is this "mass or energy of curvature"? I have never seen this anywhere. If you mean the equivalence to the stress-tensor, that is to say that the curvature of spacetime is affected by the stress tensor, not that the curvature itself has any mass or energy or whatever. $\endgroup$– VaibhavKCommented May 9 at 11:43
-
1$\begingroup$ Curvature density in FLRW is just bookkeeping, it's not actually mass/energy in the sense provided in the answer by @NíckolasAlves. It's related to spatial curvature, not spacetime curvature, and is nonzero even in an empty universe (with zero spacetime curvature). $\endgroup$– StenCommented May 9 at 19:00
|
Show 1 more comment
2 Answers
$\begingroup$
$\endgroup$
4
You can think of it as potential energy, but there is a caveat: this mass/energy is not localized. You can't say the mass is here or there, because you can't write a stress-energy-momentum tensor for the gravitational field.
However, you can measure this mass at infinity. You use, for example, the Komar integrals. These are analogues of Gauss's Law of Electrodynamics: you measure the mass inside the spacetime by looking at the behavior of the gravitational field at infinity.
Important examples of this are black holes. Black holes are vacuum solutions, so there is no matter in spacetime. Still, they have mass. All of this mass is due to the curvature of spacetime. You can't say where the mass is, but you know the spacetime has mass in the sense of Komar.
answered May 9 at 13:34
-
$\begingroup$ Is the mass due to curvature the same as the mass which formed the black hole? $\endgroup$ Commented May 9 at 16:02
-
$\begingroup$ @JohnHobson Numerically, yes (due to conservation of energy). However, there is the difference that the original mass is due to matter (for example in the form of a star) but the black hole mass is only curvature (there is no matter anywhere in the spacetime) $\endgroup$ Commented May 9 at 16:06
-
$\begingroup$ Is it possible that a black hole has a 3-sphere surface, at the event horizon, which could be capable of containing matter? $\endgroup$ Commented Jul 13 at 16:43
-
$\begingroup$ @JohnHobson Astrophysical black holes never form in an outside observer reference frame, so they indeed contain matter just outside the horizon. An eternal black hole does not. $\endgroup$ Commented Jul 13 at 23:33
$\begingroup$
$\endgroup$
Now I have heard there is an energy associated with the curvature of space. If so, it will have a mass equivalent.
Are you thinking of quantum fluctuations in the vaccum? If so, it is believed that any mass/energy created by these fluctuations averages out to zero, via the creation of anti-particle pairs. I suspect that over a large scale, the net effect of spacetime curvature would also be zero.
Keep in mind that $E=mc^2$, so it takes a LOT of energy to be equivalent to a little mass, and a LOT of mass packed into a space to start to curve spacetime in a meaningful way.
