Mechanism of gravitational acceleration in GR

[ZirkMan]

I understand how non-euclidean space curvature can cause a change in direction of inertial motion but I don't understand how it can cause acceleration.

I guess the reason will be that not only space is curved but also time and that the time curvature somehow will provide the energy needed for acceleration. Can you please explain how this mechanism of gravitational acceleration works?

 

[Dale]

Consider a rotating reference frame in flat spacetime, and look at the lines formed by the coordinates. The lines for the time coordinate form helices, they are curved. An inertially moving object will undergo coordinate acceleration in this roataing frame.

Similarly for gravity. The gravitational acceleration is caused simply by using coordinates that are curved. You can remove the acceleration simply by changing to "straight" coordinates. Of course, that is only possible over a limited region in a curved spacetime.

 

[ZirkMan]

Consider a rotating reference frame in flat spacetime, and look at the lines formed by the coordinates. The lines for the time coordinate form helices, they are curved. An inertially moving object will undergo coordinate acceleration in this roataing frame.

Similarly for gravity. The gravitational acceleration is caused simply by using coordinates that are curved. You can remove the acceleration simply by changing to "straight" coordinates. Of course, that is only possible over a limited region in a curved spacetime.

So it's the same principle as with the centripetal force only here the rotating frame is substituted with curved (non-rotating) spacetime?

 

[Dale]

So it's the same principle as with the centripetal force only here the rotating frame is substituted with curved (non-rotating) spacetime?

Yes. Even in flat spacetime you can have curved coordinates. In curved spacetime you have no choice but to use curved coordinates globally.

 

[Mentz114]

I guess the reason will be that not only space is curved but also time and that the time curvature somehow will provide the energy needed for acceleration.

Yes, that's about right. As GR relaxes into the weak field limit, Newtonian gravity is recovered and the time curvature plays a part in the potential.

Can you please explain how this mechanism of gravitational acceleration works?

No, I can't. In GR all inertial objects are moving along 4D geodesics and these will curve towards each other if gravitational attraction is present.

 

[Jonathan Scott]

I understand how non-euclidean space curvature can cause a change in direction of inertial motion but I don't understand how it can cause acceleration.

I guess the reason will be that not only space is curved but also time and that the time curvature somehow will provide the energy needed for acceleration. Can you please explain how this mechanism of gravitational acceleration works?

A minor correction; no energy is needed for gravitational acceleration. A static field causes a change in momentum, but not in energy. In Newtonian terms this is because the change in kinetic energy is balanced by a change in potential energy. In GR terms this is because the change in kinetic energy is matched by the effect on the total energy of the change in time dilation at a different potential.

 

[ZirkMan]

A minor correction; no energy is needed for gravitational acceleration.

Ok, this is exactly the point I now try to understand.

A static field causes a change in momentum, but not in energy. In Newtonian terms this is because the change in kinetic energy is balanced by a change in potential energy. In GR terms this is because the change in kinetic energy is matched by the effect on the total energy of the change in time dilation at a different potential.

Do I read it correctly that in the GR terms the increase in KE is supplied by the decreasing time potential (the higher the KE the slower the time passage at that level)?
Wouldn't this imply that the total energy of an object is higher closer to the center of attracting mass than far from it?

 

[Jonathan Scott]

Ok, this is exactly the point I now try to understand.

Do I read it correctly that in the GR terms the increase in KE is supplied by the decreasing time potential (the higher the KE the slower the time passage at that level)?
Wouldn't this imply that the total energy of an object is higher closer to the center of attracting mass than far from it?

What I mean is that if you add in the change in the KE between two locations but then multiply the resulting total energy (as in a local static frame) by the ratio of the time dilation factors at those locations (which in Newtonian terms approximately subtracts off the potential energy) then you get back to the original total energy.

The total energy for free fall in a static field is constant as seen by an observer at any fixed potential. The kinetic energy increases at lower potential, but the rest energy decreases by the same amount.

You can think of rest energy and momentum as vector quantities at right angles and of total energy as being the hypotenuse (long side) of the resulting triangle. What we call the kinetic energy is then the difference in magnitude between the total energy and the rest energy. For the sort of forces we normally encounter in Special Relativity, the rest energy is fixed and the momentum varies, so the total energy varies too. For free fall under the influence of a static gravitational field in GR, it is the magnitude of the total energy which is fixed, so when the momentum varies the rest energy varies too.

 

[ZirkMan]

What I mean is that if you add in the change in the KE between two locations but then multiply the resulting total energy (as in a local static frame) by the ratio of the time dilation factors at those locations (which in Newtonian terms approximately subtracts off the potential energy) then you get back to the original total energy.

The total energy for free fall in a static field is constant as seen by an observer at any fixed potential. The kinetic energy increases at lower potential, but the rest energy decreases by the same amount.

You can think of rest energy and momentum as vector quantities at right angles and of total energy as being the hypotenuse (long side) of the resulting triangle. What we call the kinetic energy is then the difference in magnitude between the total energy and the rest energy. For the sort of forces we normally encounter in Special Relativity, the rest energy is fixed and the momentum varies, so the total energy varies too. For free fall under the influence of a static gravitational field in GR, it is the magnitude of the total energy which is fixed, so when the momentum varies the rest energy varies too.

Ok, I understand that the total energy in a free fall has to be conserved. But when we observe the rise in KE during the fall that means that some other energy form of the falling object has to decrease (in order to conserve the total energy). The way I see it the decrease in the rest energy should be manifested as a decrease in temperature (because when observed time passes slower everything moves slower including movement of molecules and atoms and that movement = temperature).

This should be easily proven by a thought experiment where you would measure temperature of a free falling object by its EM radiation and you would record the gravitational redshift = decrease of temperature of the falling object. Is this correct?

 

[Jonathan Scott]

Ok, I understand that the total energy in a free fall has to be conserved. But when we observe the rise in KE during the fall that means that some other energy form of the falling object has to decrease (in order to conserve the total energy). The way I see it the decrease in the rest energy should be manifested as a decrease in temperature (because when observed time passes slower everything moves slower including movement of molecules and atoms and that movement = temperature).

This should be easily proven by a thought experiment where you would measure temperature of a free falling object by its EM radiation and you would record the gravitational redshift = decrease of temperature of the falling object. Is this correct?

The temperature has no special relevance here; all forms of energy which contribute to the rest energy of the object will be scaled in exactly the same way by the time dilation due to the gravitational potential. An atom in the falling object will have the same rest energy as a similar local atom which it happens to be falling past, but from the viewpoint of a fixed observer both of those atoms have less rest mass than similar atoms at a location with a higher potential.

 

[Mentz114]

Ok, I understand that the total energy in a free fall has to be conserved. But when we observe the rise in KE during the fall that means that some other energy form of the falling object has to decrease (in order to conserve the total energy). The way I see it the decrease in the rest energy should be manifested as a decrease in temperature (because when observed time passes slower everything moves slower including movement of molecules and atoms and that movement = temperature).

This should be easily proven by a thought experiment where you would measure temperature of a free falling object by its EM radiation and you would record the gravitational redshift = decrease of temperature of the falling object. Is this correct?

Kinetic energy is observer dependent. In the falling objects own rest frame it has zero KE but relative to a moving observer it may have great KE.

GR demands local conservation of energy and momentum so in a lab frame, an experiment with billiard balls colliding would still confirm conservation of energy and momentum locally.

The Newtonian concept of the gravitational field having energy does not apply. Instead we have a kinematical system, not a dynamic one.