B Conservation Laws & General Relativity: Understanding Energy

[sqljunkey]

How does general relativity shows the conservation of energy. Because I was reading and listening to something today that touched on this subject. It almost seems as though if you scale GR to larger sizes it stops working and turns into an incomplete law of nature like Newton's laws of gravitation.

 

[pervect]

There are multiple definitions of energy as a conserved global quantity in GR. The "big three" are Komar energy, which requires a stationary space-time, and Bondi and ADM energy, both of which require asymptotic flatness.

Local energy conservation of energy, in contrast to global defintions of a conserved quantity, is built into the theory. The mathematical statement of this is that the divergence of the stress-energy tensor is zero. This is rather similar to the conservation laws of fluid mechanics.

The complexities of global energy in GR have little to do (probably nothing to do) with the theory being incomplete, but more to issues with the model of space-time as a non-flat manifold.

For a popular discussion, I'd recommend "Is Energy Conserved in General Relativity", https://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html.

For a more detailed discussion, I'd recommend Wald's "General Relativity", which isgraduate level. It's not a particularly easy read, and the discussion of energy is midway through the book, so it's not something one can "pick up" on without a significant background.

 

[PeterDonis]

I was reading and listening to something today

Please give a specific reference.

 

[Ibix]

How does general relativity shows the conservation of energy.

The Einstein field equations say $G^{ab}\propto T^{ab}$. The left hand side describes curvature, and its covariant derivative, $\nabla_aG^{ab}$, is zero. This is called the Bianchi identity and is just geometry, like the sum of the angles of a triangle being 180° in Euclidean geometry. But, via the Einstein field equations, it tells us something physical - that $\nabla_aT^{ab}=0$, which is the law of local conservation of stress-energy that @pervect mentioned. Roughly speaking, it says that the difference between the stress energy in a small region of space now and a tiny time later is equal to the amount that flowed in or out of the region in that time - or, to put it another way, that stress-energy is neither created nor destroyed. All GR spacetimes respect this. They cannot avoid it.

The problem, also as pervect says, is that you can't generally take that local law and turn it into a global one (with honourable exceptions). I have the impression that opinion is divided over whether this is a problem and whether proposed solutions (if, indeed, it is a problem) are generally legitimate.

 

[etotheipi]

For any Killing vector field $\xi$ there is a Komar charge $Q_{\xi}(V) \sim \int_{\partial V} \star d\xi$ which is time-independent provided the corresponding current $J_a \sim \left(T_{ab} - \frac{1}{2}Tg_{ab} \right)\xi^b$ vanishes on $\partial V$. If the spacetime is stationary then by definition there's a timelike Killing field $k^a$ giving rise to the Komar energy $M_{\mathrm{K}} = - \dfrac{1}{8\pi} \lim_{r \rightarrow \infty} \int_{S_r^2} \star dk$.

If the spacetime is non-stationary then you can still define energy as the value of the Hamiltonian evaluated on a solution with an asymptotically flat end, and this gives rise to the ADM energy $E_{\mathrm{ADM}} = \dfrac{1}{16\pi} \lim_{r\rightarrow \infty} \int_{S_r^2} dA \, n_i (\partial_j h_{ij} - \partial_i h_{ji})$. You can actually also show that $M_{\mathrm{K}} = E_{\mathrm{ADM}}$ if the spacetime is both stationary and asymptotically flat.

 

[vanhees71]

The Einstein field equations say $G^{ab}\propto T^{ab}$. The left hand side describes curvature and its covariant derivative, $\nabla_aG^{ab}$, is zero. This is called the Bianchi identity and is just geometry, like the sum of the angles of a triangle being 180° in Euclidean geometry. But it means that $\nabla_aT^{ab}=0$, which is the law of local conservation of stress-energy that @pervect mentioned. Roughly speaking, it says that the difference in stress energy in a region of space now and a tiny time later is equal to the amount that flowed out of the region in that time - or, stress-energy is neither created nor destroyed. All GR spacetimes respect this - they cannot avoid it.

The problem, also as pervect says, is that you can't generally take that local law and turn it into a global one (with honourable exceptions). I have the impression that opinion is divided over whether this is a problem and whether proposed solutions (if, indeed, it is a problem) are generally legitimate.

You can also interpret the local conservation of energy as an integrability condition for the Einstein equations with the energy-momentum-stress tensor of "matter and radiation" as sources of the gravitational field. Mathematically it originates from the gauge symmetry of GR, which can be derived from SR by gauging the
Poincare symmetry of special-relativistic spacetime.

You have the same phenomenon in classical electrodynamics: Due to gauge invariance the (local) conservation of electric charge is an integrability condition for Maxwell's equations.

If you use as a model for the sources some continuum-mechanical system the local energy-momentum conservation law $\nabla_{\mu} T^{\mu \nu}=0$ describes the complete dynamics of the continuum mechanical system, i.e., the equations of motion is automatically solved when you find a solution of Einstein's field equations for this system.

The most simple examples are dust with $T^{\mu \nu}=\mu c^2 u^{\mu} u^{\nu}$, where $\mu$ is the mass density of the dust (as measured in the local rest frame) and $u^{\mu}$ it's four-velocity field (fulfilling the constraint $g_{\mu \nu} u^{\mu} u^{\nu}=1$ (west-coast convention and using the "normalized" version of the four-velocity, i.e., in SR $(u^{\mu})=\gamma(1,\vec{\beta})$ with $\vec{\beta}=\vec{v}/c$), and an ideal fluid with $T^{\mu \nu}=h u^{\mu} u^{\nu}-P g^{\mu \nu}$, where $h=\epsilon+P$ is the enthalpy density, as measured in the local restframe of the fluid cell and $P$ the pressure.

For dust you can easily show that the equation of motion $\nabla_{\mu} T^{\mu \nu}=0$ leads to the geodesic equation for $u^{\mu}$ as expected for non-interacting particles in a gravitational field.