Is a Metric Feedback Possible in General Relativity?

[OccamsRazor]

Apologies for not doing too much research prior to asking this question; I suppose actually delving into the mathematics would reveal the answer I'm looking for but I haven't taken the time just yet.

Considering the concept of GR where matter/energy tells space how to curve and space tells how matter to move, how do we avoid the feedback that would be cause by space telling matter to move faster which in turn increases the energy/matter of moving body? (in such a case wouldn't the metric then be a function of itself) ?

could such a metric exist? or no? or could it exist but simply not satisfy the field equations?

 

[bcrowell]

Considering the concept of GR where matter/energy tells space how to curve and space tells how matter to move, how do we avoid the feedback that would be cause by space telling matter to move faster which in turn increases the energy/matter of moving body?

In nontechnical language, energy is conserved, so there can't be any such increase.

One can make this statement more mathematically precise, but your question is posed in nonmathematical language. If you wanted to get into a more precise statement of this, you would want to start by recognizing that the source of gravitational fields in GR is not energy but the stress-energy tensor. The stress-energy tensor has zero divergence, which means that energy-momentum is locally conserved.

 

[OccamsRazor]

In nontechnical language, energy is conserved, so there can't be any such increase. (One can make this statement more mathematically precise, but your question is posed in nonmathematical language.)

So, in nontechnical language, would the constraint of energy conservation cause space to bend in a way that will compensate for any kinetic energy increase? Is there energy associated with curved space (intrinsic to the space, and not merely a passive player that tells matter how to move in a more energetic way? )

 

[bcrowell]

So, in nontechnical language, would the constraint of energy conservation cause space to bend in a way that will compensate for any kinetic energy increase? Is there energy associated with curved space (intrinsic to the space, and not merely a passive player that tells matter how to move in a more energetic way? )

In nontechnical language, the answer to both questions is yes.

The technicalities limit the precision with which we can define where the energy-momentum is localized, and they also limit our ability to define its conservation globally rather than locally. GR also doesn't allow us to isolate kinetic energy and potential energy as separate phenomena, or likewise energy and momentum.

 

[OccamsRazor]

when you say the technicalities limit our ability to define where energy-momentum is localized, do you mean technicalities in the theoretical framework of GR? I figured since it was a classical theory that things 'exactness' were capable.

and even if kinetic and potential energies (or energy and momentum) get mixed up in the space-time view of a moving body, shouldn't we still be able to make a clear demarcation between the energy-momentum of a body from the energy-momentum of curved space-time? Or did you mean something else ?

 

[bcrowell]

when you say the technicalities limit our ability to define where energy-momentum is localized, do you mean technicalities in the theoretical framework of GR? I figured since it was a classical theory that things 'exactness' were capable.

This might be a better topic for a separate thread, but here's the basic idea. When we define an energy density for a field such as the electric field, it's proportional to ##E^2##. The analogous thing for the Newtonian gravitational field is ##g^2##, where g is the freshman physics quantity that equals 9.8 m/s2 on earth. But by the equivalence principle, we can always pick coordinates such that, at some chosen point, g=0. For example, g=0 in my living room, if I use the coordinates of a free-falling observer. Therefore we can't unambiguously say whether there is gravitational energy in my living room.

and even if kinetic and potential energies (or energy and momentum) get mixed up in the space-time view of a moving body, shouldn't we still be able to make a clear demarcation between the energy-momentum of a body from the energy-momentum of curved space-time?

This might also be a better topic for a separate thread. There is a nice discussion of this in Exploring Black Holes: Introduction to General Relativity, by Taylor and Wheeler.

 

[OccamsRazor]

I actually do have that book, do you have a chapter name or page number that addresses this question?

Would you suggest making the thread simply of the second question ?

 

[bcrowell]

I actually do have that book, do you have a chapter name or page number that addresses this question?

I don't have the book handy, but they have a discussion of the world-line of a test particle, and they go into the question of why you can't separate the PE from the KE.

 

[cosmik debris]

Apologies for not doing too much research prior to asking this question; I suppose actually delving into the mathematics would reveal the answer I'm looking for but I haven't taken the time just yet.

Considering the concept of GR where matter/energy tells space how to curve and space tells how matter to move, how do we avoid the feedback that would be cause by space telling matter to move faster which in turn increases the energy/matter of moving body? (in such a case wouldn't the metric then be a function of itself) ?

could such a metric exist? or no? or could it exist but simply not satisfy the field equations?

This is a very non-technical explanation. I think that this does happen to a certain extent because the field equations are non-linear, but the increase in curvature caused by the increasing KE of the test body diminishes rapidly until steady state is achieved.