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S10h. Energy in general relativity
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Energy is no absolute concept, but depends on the observer
(in the nonrelativistic case, by choice of a velocity,
in the relativistic case, by choice of time-like unit
vector that defines the direction of time and hence the
time coordinate).
In classical mechanics there is always a (up to rotations)
distinguished center of mass frame where the whole system
is at rest and the center of mass at zero.
The observer is usually (silently) considered to be at rest
with respect to that frame; then there is no ambiguity
left in the energy.
In special relativity things are already more problematic
since there is no natural center of mass. But one can fix
the time direction by taking it to be that of the total
4-momentum of the whole system. This again fixes a frame,
now up to Euclidean motions. On the other hand, this is not
what an observer (who has a slightly different eigentime
depending on its 4-momentum) sees, and must be corrected
accordingly.
In general relativity the conserved total 4-momentum is
identically zero, so there is no longer a way to fix a
time direction. But assuming an asymptotically flat
space-time one can take its flat coordinate system
(determined up to a Poincare transformation) and
use it to chart the localized part, and gets a Minkowski
description, to which the preceding applies.
In general relativity, the concept of energy depends on the
choice of a spacelike hypersurface defining a region of space
and a time-like vector field along that hypersurface defining
the direction of time: Then the integral of [part of]
the (0,0)-component of the energy-momentum tensor over this
hypersurface defines the corresponding [part of the]
energy in this region.
This allows one to talk about the (observer-dependent) energy
of a subsystem, or of all matter in the universe, etc.
Observer-independent is the energy-momentum tensor density
as a whole, but not energy.
The weak-field limit defines a preferred coordinate system,
thus reducing the arbitrariness to the choice of the time
direction, and the nonrelativistic limit fixes this choice
to be the direction of the total momentum of the reference
object (e.g., the earth or sun or our galaxy). This makes
everything completely determined and gives us a good
energy for everyday life.
Note that using the concept of energy does not require
a global conservation law.
Even in nonrelativistic classical mechanics, energy is conserved
only for isolated systems, while the concept is used very
profitably in all sorts of nonisolated settings. It just means that
one needs to account in the balance equations for what happens
at the boundary, and (if necessary) include friction terms
(which describe, so to speak, the boundary to the neglected
microscopic degrees of freedom).
Thus, to connect general relativity to what most physicists actually
study, namely systems localized in a small region of space and time
(small may mean, e.g., a laboratory, the earth, the solar system,
or our galaxy - within an hour, a year, a few millenia, etc.)
one needs to make precise what energy means for such pieces of the
whole universe.
This requires that the observer specifies the region of space of
interest, and the length of time of interest, including the way time
is supposed to flow. The observer also has to specify which part of
the energy is of interest, i.e., the terms in the energy-momentum
tensor that define the system (contrasted to the environment -
which make up all the other terms).
After all that is done, energy has a well-defined meaning,
as given above.
On the other hand, the observer-independent notion generalizing
energy is the full energy-momentum tensor; its tensor
nature reflects the need for observer information to extract
from it numerical values, i.e. real numbers that can be compared
with experiment. But apart from energy it also contains the
observer-independent part of the information about momentum
and stress, which themselves are also observer-dependent.