[JS]: Of course it is. That's the way it is even in special relativity!
Energy density is the 00 component of Tuv it's not a frame invariant you
damn fool.
You do not seem to understand tensors - how they transform.
Here Xu^u' is the local T4 frame transformation A to B (locally coincident
and generally LNIF).
all the tensor components in Frame A contribute to the energy density in
Frame B in the general case.
[PZ]: It's not that the universe is stranger than I can imagine -- it's that since Einstein published his 1905 relativity paper,
physicists have completely *lost their marbles* and can no longer distinguish between subjective appearances and objective physical reality, or between actual physics and naive-empiricist *applied math*.
[JS]: In the case of quantum theory
and
string theory you are perhaps correct, but not in this case of classical
general relativity and its extension to include torsion and possibly conformal
GMD fields in addition to the 1916 curvature field.
[PZ]: I agree with Einstein: "A good joke should not be repeated too often".
Lensman wrote: What does he mean by "The observer dependence of the quasi-local energy (QLE) and momentum in the Schwarzschild geometry"?
http://arxiv.org/abs/0801.3683 [JS]: Here is what John Baez writes in:
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
Is Energy Conserved in General Relativity?
In special cases, yes. In general -- it depends on what
you mean by "energy", and what you mean by
"conserved".
In flat spacetime (the backdrop for special relativity)
you can phrase energy conservation in two ways: as a differential equation, or
as an equation involving integrals (gory details below). The two formulations
are mathematically equivalent. But when you try to generalize this to curved
spacetimes (the arena for general relativity) this equivalence breaks down.
The differential form extends with nary a hiccup; not so
the integral form.
The
differential form says, loosely speaking, that no energy is created in any
infinitesimal piece of spacetime. The integral form says the same for a
finite-sized piece. (This may remind you of the "divergence" and "flux"
forms of Gauss's law in electrostatics, or the equation of continuity in fluid
dynamics. Hold on to that thought!)
An
infinitesimal piece of spacetime "looks flat", while the effects of curvature
become evident in a finite piece. (The same holds for curved surfaces in
space, of course). GR relates
curvature to gravity. Now,
even in Newtonian physics, you must include gravitational potential energy to
get energy conservation. And GR introduces the new phenomenon of gravitational
waves; perhaps these carry energy as well? Perhaps we need to include
gravitational energy in some fashion, to arrive at a law of energy conservation
for finite pieces of spacetime?
Casting about
for a mathematical expression of these ideas, physicists came up with something
called an energy pseudo-tensor. (In fact, several of 'em!) Now, GR takes pride in treating all coordinate
systems equally. Mathematicians invented tensors precisely to meet this
sort of demand -- if a tensor
equation holds in one coordinate system, it holds in all. Pseudo-tensors are not tensors
(surprise!), and this alone raises eyebrows in some circles. In GR, one must always guard against
mistaking artifacts of a particular coordinate system for real physical effects.
(See the FAQ entry on black holes for some
examples.)
These
pseudo-tensors have some rather strange properties. If you choose the "wrong"
coordinates, they are non-zero even in flat empty spacetime. By another choice
of coordinates, they can be made zero at any chosen point, even in a spacetime
full of gravitational radiation. For these reasons, most physicists who work in
general relativity do not believe the pseudo-tensors give a good
local definition of energy density, although
their integrals are sometimes useful as a measure of total
energy.
[JS]:
Pass now to the general case of any spacetime satisfying
Einstein's field equation. It is easy to generalize the differential form of
energy-momentum conservation, Equation 2:
Equation 3: (valid in any GR
spacetime)
covariant_div(T) = sum_mu nabla_mu(T) = 0
(where nabla_mu = covariant
derivative)
(Side comment: Equation 3 is the correct generalization
of Equation 1 for SR when non-Minkowskian coordinates are
used.)
GR relies heavily on the covariant derivative, because
the covariant derivative of a tensor is a tensor, and as we've seen, GR loves
tensors. Equation 3 follows from
Einstein's field equation (because something called Bianchi's identity
says that covariant_div(G)=0). But Equation 3 is no longer equivalent to
Equation 1!
Why not? Well, the familiar form of Gauss's theorem
(from electrostatics) holds for any spacetime, because essentially you are
summing fluxes over a partition of the volume into infinitesimally small pieces.
The sum over the faces of one infinitesimal piece is a divergence. But the total
contribution from an interior face is zero, since what flows out of one piece
flows into its neighbor. So the integral of the divergence over the volume
equals the flux through the boundary. "QED".
But for the
equivalence of Equations 1 and 3, we would need an extension of Gauss's theorem.
Now the flux through a face is not a scalar, but a vector (the flux of
energy-momentum through the face). The argument just sketched involves adding
these vectors, which are defined at different points in spacetime. Such "remote
vector comparison" runs into trouble precisely for curved
spacetimes.
The
mathematician Levi-Civita invented the standard solution to this problem, and
dubbed it "parallel transport". It's easy to picture parallel transport: just move the vector
along a path, keeping its direction "as constant as possible".
(Naturally, some non-trivial mathematics lurks behind the phrase in quotation
marks. But even pop-science expositions of GR do a good job explaining parallel
transport.) The parallel transport of
a vector depends on the transportation path; for the canonical example,
imagine parallel transporting a vector on a sphere. But parallel transportation
over an "infinitesimal distance" suffers no such ambiguity. (It's not hard to
see the connection with curvature.)
To compute a
divergence, we need to compare quantities (here vectors) on opposite faces.
Using parallel transport for this leads to the covariant divergence. This is
well-defined, because we're dealing with an infinitesimal hypervolume. But to
add up fluxes all over a finite-sized hypervolume (as in the contemplated
extension of Gauss's theorem) runs smack into the dependence on transportation
path. So the flux integral is not well-defined, and we have no analogue for
Gauss's theorem.
One way to
get round this is to pick one coordinate system, and transport vectors so their
components stay constant. Partial derivatives
replace covariant derivatives, and Gauss's theorem is restored. The energy pseudo-tensors take this
approach (at least some of them do). If you can mangle Equation 3
(covariant_div(T) = 0) into the form:
coord_div(Theta) = 0
then you can
get an "energy conservation law" in integral form. Einstein was the first
to do this; Dirac, Landau and Lifshitz, and Weinberg all came up with variations
on this theme. We've said enough already on the pros and cons of this
approach.
We will not delve into definitions of energy in general
relativity such as the Hamiltonian (amusingly, the energy of a closed universe always works out to zero according to
this definition), various kinds of energy one hopes to obtain by
"deparametrizing" Einstein's equations, or "quasilocal energy". There's quite a bit to say about this
sort of thing! Indeed, the issue of
energy in general relativity has a lot to do with the notorious "problem of
time" in quantum gravity. . . but that's another can of
worms.
References (vaguely in order of
difficulty):
- Clifford Will, The renaissance of general
relativity, in The New Physics (ed. Paul Davies) gives a semi-technical discussion of
the controversy over gravitational radiation.
- Wheeler, A Journey into
Gravity and Spacetime. Wheeler's try at a
"pop-science" treatment of GR. Chapters 6 and 7 are a tour-de-force: Wheeler
tries for a non-technical explanation of Cartan's formulation of Einstein's field equation. It might
be easier just to read MTW!)
- Taylor and Wheeler, Spacetime Physics.
- Goldstein, Classical
Mechanics.
- Arnold,
Mathematical Methods in Classical
Mechanics.
- Misner, Thorne, and Wheeler (MTW),
Gravitation, chapters 7, 20, and 25
- Wald,
General Relativity, Appendix E.
This has the Hamiltonian formalism and a bit about deparametrizing, and chapter
11 discusses energy in asymptotically flat
spacetimes.
- H. A. Buchdahl, Seventeen Simple Lectures on General Relativity
Theory Lecture 15 derives the
energy-loss formula for the binary star, and criticizes the
derivation.
- Sachs and Wu, General Relativity for
Mathematicians, chapter 3.
- John Stewart,
Advanced General Relativity. Chapter 3
(Asymptopia) shows just how careful one has to be in
asymptotically flat spacetimes to recover energy conservation. Stewart also
discusses the Bondi-Sachs mass, another contender for
"energy".
- Damour, in 300 Years of
Gravitation (ed. Hawking and Israel). Damour
heads the "Paris group", which has been active in the theory of gravitational
radiation.
- Penrose and
Rindler, Spinors and Spacetime, vol II,
chapter 9. The Bondi-Sachs mass generalized.
- J. David Brown and James York Jr.,
Quasilocal energy in general
relativity, in Mathematical Aspects of Classical Field
Theory.
