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Energy in General Relativity

Energy conservation arises due to invariance under translations in time, and in general this will not hold. In general relativity, we do have the analogue,

$$\nabla_\mu T^{\mu\nu} = 0$$

however this does not imply energy is conserved, because one cannot bring the expression into an integral form, as one can normally when applying Noether's theorem to field theories, where we could define a conserved current $\partial_\mu j^\mu = 0$ and a conserved charge,

$$Q = \int d^3x \, j^0.$$

Locality

It is possible to define a Landau-Lifshitz pseudo-tensor $\tau_{\mu\nu}$ which ascribes stress-energy to the gravitational field, such that,

$$\partial_\mu (T^{\mu\nu} + \tau^{\mu\nu}) = 0,$$

from which one can define momentum $P^\mu$ and angular momentum $J^{\mu\nu}$. However, the modified stress-energy and $\tau_{\mu\nu}$ itself has no geometric, coordinate free significance. It may vanish in one coordinate system and not in another.

The trouble boils down to the fact that gravitational energy cannot be localised. For electromagnetism, one can speak of a region of space-time with some energy density due to the electromagnetic field, which is responsible for inducing curvature and changing the wordlines passing through it.

Due to the equivalence principle, locally we can always define a coordinate system wherein the gravitational field vanishes, which means it does not make sense to speak of a local gravitational energy density.

Alternate Definitions

Nevertheless, there are further analogues of energy and other quantities of the Hamiltonian formalism in general relativity which are sometimes useful, though there are issues with these as aforementioned, such as either coordinate dependence, or other ambiguities. One expression is the quasilocal energy, defined as,

$$E = -\int_B d^2 x \, \frac{\delta S_{\mathrm{cl}}}{\delta N} = \frac{1}{\kappa}\int_B d^2x\, \sqrt{\sigma}k \big\rvert_{\mathrm{cl}}$$

where $B$ is the boundary of a spatial hypersurface $\Sigma$, with $\sigma$ and $k$ the metric and extrinsic curvature respectively. Contributions from flat space must be subtracted off typically, and there is an ambiguity in this due to two possible signs of the normal.

If we parametrize a system by a coordinate $\lambda$ for a path in the state space, then the action of a system is, $$S = \int_{\lambda'}^{\lambda''}d\lambda \, \left( p\dot x - \dot t H(x,p,t)\right).$$

For a classical history,

$$H_{\mathrm{cl}}\big\rvert_{\lambda''} = -\frac{\delta S_{\mathrm{cl}}}{\delta t''}$$

which is to say the energy at the boundary $\lambda''$ is minus the change in the classical action due to an increase in the final boundary time, $t(\lambda'') = t''.$ The expression for the quasilocal energy in general relativity is precisely the closest analogue to this Hamilton-Jacobi equation.


 

For more details and an illuminating discussion of energy conservation in general relativity, see Quasilocal Energy in General Relativity by D. Brown and J.W. York in the text, Mathematical Aspects of Classical Field Theory 132 by the AMS.

Energy conservation arises due to invariance under translations in time, and in general this will not hold. In general relativity, we do have the analogue,

$$\nabla_\mu T^{\mu\nu} = 0$$

however this does not imply energy is conserved, because one cannot bring the expression into an integral form, as one can normally when applying Noether's theorem to field theories, where we could define a conserved current $\partial_\mu j^\mu = 0$ and a conserved charge,

$$Q = \int d^3x \, j^0.$$

Nevertheless, there are analogues of energy and other quantities of the Hamiltonian formalism in general relativity, though there are issues with these such as either coordinate dependence, or other ambiguities. One expression is the quasilocal energy, defined as,

$$E = -\int_B d^2 x \, \frac{\delta S_{\mathrm{cl}}}{\delta N} = \frac{1}{\kappa}\int_B d^2x\, \sqrt{\sigma}k \big\rvert_{\mathrm{cl}}$$

where $B$ is the boundary of a spatial hypersurface $\Sigma$, with $\sigma$ and $k$ the metric and extrinsic curvature respectively. Contributions from flat space must be subtracted off typically, and there is an ambiguity in this due to two possible signs of the normal.

If we parametrize a system by a coordinate $\lambda$ for a path in the state space, then the action of a system is, $$S = \int_{\lambda'}^{\lambda''}d\lambda \, \left( p\dot x - \dot t H(x,p,t)\right).$$

For a classical history,

$$H_{\mathrm{cl}}\big\rvert_{\lambda''} = -\frac{\delta S_{\mathrm{cl}}}{\delta t''}$$

which is to say the energy at the boundary $\lambda''$ is minus the change in the classical action due to an increase in the final boundary time, $t(\lambda'') = t''.$ The expression for the quasilocal energy in general relativity is precisely the closest analogue to this Hamilton-Jacobi equation.


 

For more details and an illuminating discussion of energy conservation in general relativity, see Quasilocal Energy in General Relativity by D. Brown and J.W. York in the text, Mathematical Aspects of Classical Field Theory 132 by the AMS.

Energy in General Relativity

Energy conservation arises due to invariance under translations in time, and in general this will not hold. In general relativity, we do have the analogue,

$$\nabla_\mu T^{\mu\nu} = 0$$

however this does not imply energy is conserved, because one cannot bring the expression into an integral form, as one can normally when applying Noether's theorem to field theories, where we could define a conserved current $\partial_\mu j^\mu = 0$ and a conserved charge,

$$Q = \int d^3x \, j^0.$$

Locality

It is possible to define a Landau-Lifshitz pseudo-tensor $\tau_{\mu\nu}$ which ascribes stress-energy to the gravitational field, such that,

$$\partial_\mu (T^{\mu\nu} + \tau^{\mu\nu}) = 0,$$

from which one can define momentum $P^\mu$ and angular momentum $J^{\mu\nu}$. However, the modified stress-energy and $\tau_{\mu\nu}$ itself has no geometric, coordinate free significance. It may vanish in one coordinate system and not in another.

The trouble boils down to the fact that gravitational energy cannot be localised. For electromagnetism, one can speak of a region of space-time with some energy density due to the electromagnetic field, which is responsible for inducing curvature and changing the wordlines passing through it.

Due to the equivalence principle, locally we can always define a coordinate system wherein the gravitational field vanishes, which means it does not make sense to speak of a local gravitational energy density.

Alternate Definitions

Nevertheless, there are further analogues of energy and other quantities of the Hamiltonian formalism in general relativity which are sometimes useful, though there are issues with these as aforementioned, such as either coordinate dependence, or other ambiguities. One expression is the quasilocal energy, defined as,

$$E = -\int_B d^2 x \, \frac{\delta S_{\mathrm{cl}}}{\delta N} = \frac{1}{\kappa}\int_B d^2x\, \sqrt{\sigma}k \big\rvert_{\mathrm{cl}}$$

where $B$ is the boundary of a spatial hypersurface $\Sigma$, with $\sigma$ and $k$ the metric and extrinsic curvature respectively. Contributions from flat space must be subtracted off typically, and there is an ambiguity in this due to two possible signs of the normal.

If we parametrize a system by a coordinate $\lambda$ for a path in the state space, then the action of a system is, $$S = \int_{\lambda'}^{\lambda''}d\lambda \, \left( p\dot x - \dot t H(x,p,t)\right).$$

For a classical history,

$$H_{\mathrm{cl}}\big\rvert_{\lambda''} = -\frac{\delta S_{\mathrm{cl}}}{\delta t''}$$

which is to say the energy at the boundary $\lambda''$ is minus the change in the classical action due to an increase in the final boundary time, $t(\lambda'') = t''.$ The expression for the quasilocal energy in general relativity is precisely the closest analogue to this Hamilton-Jacobi equation.

For more details and an illuminating discussion of energy conservation in general relativity, see Quasilocal Energy in General Relativity by D. Brown and J.W. York in the text, Mathematical Aspects of Classical Field Theory 132 by the AMS.

Source Link
JamalS
  • 19.5k
  • 6
  • 59
  • 107

Energy conservation arises due to invariance under translations in time, and in general this will not hold. In general relativity, we do have the analogue,

$$\nabla_\mu T^{\mu\nu} = 0$$

however this does not imply energy is conserved, because one cannot bring the expression into an integral form, as one can normally when applying Noether's theorem to field theories, where we could define a conserved current $\partial_\mu j^\mu = 0$ and a conserved charge,

$$Q = \int d^3x \, j^0.$$

Nevertheless, there are analogues of energy and other quantities of the Hamiltonian formalism in general relativity, though there are issues with these such as either coordinate dependence, or other ambiguities. One expression is the quasilocal energy, defined as,

$$E = -\int_B d^2 x \, \frac{\delta S_{\mathrm{cl}}}{\delta N} = \frac{1}{\kappa}\int_B d^2x\, \sqrt{\sigma}k \big\rvert_{\mathrm{cl}}$$

where $B$ is the boundary of a spatial hypersurface $\Sigma$, with $\sigma$ and $k$ the metric and extrinsic curvature respectively. Contributions from flat space must be subtracted off typically, and there is an ambiguity in this due to two possible signs of the normal.

If we parametrize a system by a coordinate $\lambda$ for a path in the state space, then the action of a system is, $$S = \int_{\lambda'}^{\lambda''}d\lambda \, \left( p\dot x - \dot t H(x,p,t)\right).$$

For a classical history,

$$H_{\mathrm{cl}}\big\rvert_{\lambda''} = -\frac{\delta S_{\mathrm{cl}}}{\delta t''}$$

which is to say the energy at the boundary $\lambda''$ is minus the change in the classical action due to an increase in the final boundary time, $t(\lambda'') = t''.$ The expression for the quasilocal energy in general relativity is precisely the closest analogue to this Hamilton-Jacobi equation.


For more details and an illuminating discussion of energy conservation in general relativity, see Quasilocal Energy in General Relativity by D. Brown and J.W. York in the text, Mathematical Aspects of Classical Field Theory 132 by the AMS.