# Does GR predict SR mass changes?

GR can be used to show that the predicted time dilation on the surface of a planet equals the time dilation required by SR at that planet's escape velocity. Can it also be used to independently calculate the mass increase at that velocity without any reference to SR?

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Nugatory
Mentor
GR can be used to show that the predicted time dilation on the surface of a planet equals the time dilation required by SR at that planet's escape velocity. Can it also be used to independently calculate the mass increase at that velocity without any reference to SR?
No, and it's unlikely that such a derivation could be found because the derivation of the relativistic mass equation is done in a local inertial frame - and GR doesn't tell us anything about local inertial frames that SR hasn't already told us.

This is in contrast to gravitational time dilation, which is pretty clearly a non-local phenomenon. Even then, it's more of a happy accident than a deep result that the two time dilations you mention happen to come out the same.

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pervect
Staff Emeritus
GR is built on top of SR - it basically adds the idea that space-time is curved and the field equations that describe the evolution of said curvature to special relativity. So it's not really possible to cut off GR from SR in the manner in which you suggest.

Additionally, the concept of "relativistic mass" you describe isn't used in GR, and it's hardly used in modern treatments of SR anymore. (You'll find it in old textbooks, and it also seems to be widely used in popularizations).

The treatment of mass in GR is rather complex, there are at least three different sorts of mass that are in common use, none of which is universally applicable. The so-called Komar mass (also called energy at infinity) which applies to static situations, is probably the one most closely related to your observation about escape velocity. Other sorts of mass in common use in GR are the ADM and Bondi masses.

Can it also be used to independently calculate the mass increase at that velocity without any reference to SR?
Yes, GR can be used to calculate the change of gravitational mass, that SR does not refers to:

http://home.comcast.net/~peter.m.brown/ref/mass_articles/Olson_Guarino_1985.pdf [Broken]

The usefulness of such calculations is another question.

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PeterDonis
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2019 Award
This is in contrast to gravitational time dilation, which is pretty clearly a non-local phenomenon.
So is escape velocity. That's not a coincidence; see below.

Even then, it's more of a happy accident than a deep result that the two time dilations you mention happen to come out the same.
It's not an accident. Consider:

* Escape velocity is a measure of the energy per unit mass that needs to be added to an object to get it from rest at the surface of the planet to infinity--i.e., it's a measure of the change in energy per unit mass from the surface of the planet to infinity. But mass and energy are the same thing in relativity, so it's really a measure of the change in energy per unit of energy, i.e., the "fractional change in energy" from the surface of the planet to infinity.

* Gravitational time dilation is a measure of the redshift of light emitted from the surface of the planet and received at infinity--i.e., it's a measure of the change in energy per photon from the surface of the planet to infinity. But the redshift is really a ratio; it's the change in energy per unit of energy in the photon, i.e.,...the "fractional change in energy" from the surface of the planet to infinity.

Another way of seeing the correspondence is to imagine the following thought experiment:

* A device at infinity creates a bunch of photons with total energy ##E## and sends them down to the surface of the planet.

* At the planet's surface, the photons are converted to an object with rest mass ##m = E## as measured locally. The extra energy in the photons (since they were blueshifted during the descent) is used to boost the object to escape velocity and send it back out to infinity.

* When the object reaches infinity, it is at rest (since it had just enough velocity to escape), so its total energy is equal to its rest mass ##m##, which equals ##E##.

Obviously, the energy gained by the photons in falling (which is determined by the gravitational redshift/time dilation) must be equal to the energy needed to boost the object to escape velocity; otherwise conservation of energy is violated.