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Let us begin with a Lagrangian of the form

$$\mathscr L= \frac 12 \sqrt{-g}g^{\mu\nu}\partial_\mu\phi(x)\partial_\nu\phi(x)+\mathscr L_g,$$

where $$\mathscr L_g=\frac 1{16\pi k}\sqrt{-g}R.$$ Suppose as well that there are no Killing vectors associated to the metric $g^{\mu\nu}$ except for, say, a global timelike Killing vector if it helps the argument.

Associated to $\mathscr L$ is a locally conserved set of 10 currents from the Poincare group:

$$T^{\mu\nu}=\partial^\mu\phi(x)\partial^\nu\phi(x)-g^{\mu\nu}\mathscr L$$

for each spacetime translation, and $\epsilon_{\alpha\beta}x^\alpha T^{\mu\beta}$ for each spacetime rotation.

Locally we have $$\nabla_\mu T^{\mu\nu}=0$$ so these quantities are conserved only locally.

My question is, what is the obstacle to patching these locally conserved quantities together to make a globally conserved quantity:

$$Q=\int T^{0 \nu}f_\nu \;d^3x$$

with

$$dQ/dt=0$$

where $f_\nu$ might be a gluing function connecting the momentum flowing out of one patch of infinitesimal volume and into another?

(Edit: I realize there may not be a tensor associated to this conserved quantity but even a pseudo tensor involving only the fields would be satisfying, if it exists. So for example, to get the ball rolling, we can start with an object of the form

$M^{\lambda\mu\nu}=\frac 12\int_{a^\mu}^{x^\lambda}ds T^{\mu\nu}(s)-\frac 12\int_{a^\mu}^{x^\mu}ds T^{\lambda\nu}(s)$,

and then set $t^{\mu\nu}=\partial_\lambda M^{\lambda\mu\nu}$.

$t^{\mu\nu}$ is a psuedo tensor that is conserved $\partial_\mu t^{\mu\nu}=0$ generically by the antisymmetry in $\lambda,\ \mu$, and as well

$t^{\mu\nu}=T^{\mu\nu}(x)+\frac 12\int^{x^\mu}ds\ \partial_\lambda T^{\lambda\nu}(s)+\frac 12\delta^{\mu}_\lambda T^{\lambda\nu}(x|_{x^\mu=a^\mu})$

In flat space this quantity is almost conserved up to the boundary term $\frac 12\delta^{\mu}_\lambda T^{\lambda\nu}(x|_{x^\mu=a^\mu})$ where $\delta^\mu_\lambda$ is a Kronecker delta.

Of course, the boundary term ruins it. That, and the lack of symmetry in mu and nu, but this should give the idea of what could with a better choice of starting point.)

Let us begin with a Lagrangian of the form

$$\mathscr L= \frac 12 \sqrt{-g}g^{\mu\nu}\partial_\mu\phi(x)\partial_\nu\phi(x)+\mathscr L_g,$$

where $$\mathscr L_g=\frac 1{16\pi k}\sqrt{-g}R.$$ Suppose as well that there are no Killing vectors associated to the metric $g^{\mu\nu}$ except for, say, a global timelike Killing vector if it helps the argument.

Associated to $\mathscr L$ is a locally conserved set of 10 currents from the Poincare group:

$$T^{\mu\nu}=\partial^\mu\phi(x)\partial^\nu\phi(x)-g^{\mu\nu}\mathscr L$$

for each spacetime translation, and $\epsilon_{\alpha\beta}x^\alpha T^{\mu\beta}$ for each spacetime rotation.

Locally we have $$\nabla_\mu T^{\mu\nu}=0$$ so these quantities are conserved only locally.

My question is, what is the obstacle to patching these locally conserved quantities together to make a globally conserved quantity:

$$Q=\int T^{0 \nu}f_\nu \;d^3x$$

with

$$dQ/dt=0$$

where $f_\nu$ might be a gluing function connecting the momentum flowing out of one patch of infinitesimal volume and into another?

(Edit: I realize there may not be a tensor associated to this conserved quantity but even a pseudo tensor involving only the fields would be satisfying, if it exists. So for example, to get the ball rolling, we can start with an object of the form

$M^{\lambda\mu\nu}=\frac 12\int_{a^\mu}^{x^\lambda}ds T^{\mu\nu}(s)-\frac 12\int_{a^\mu}^{x^\mu}ds T^{\lambda\nu}(s)$,

and then set $t^{\mu\nu}=\partial_\lambda M^{\lambda\mu\nu}$.

$t^{\mu\nu}$ is a psuedo tensor that is conserved $\partial_\mu t^{\mu\nu}=0$ generically by the antisymmetry in $\lambda,\ \mu$. Thus,

$t^{\mu\nu}=T^{\mu\nu}(x)+\frac 12\int^{x^\mu}ds\ \partial_\lambda T^{\lambda\nu}(s)+\frac 12\delta^{\mu}_\lambda T^{\lambda\nu}(x|_{x^\mu=a^\mu}).$

In flat space this quantity is almost the tensor we are looking for up to the boundary term $\frac 12\delta^{\mu}_\lambda T^{\lambda\nu}(x|_{x^\mu=a^\mu})$ where $\delta^\mu_\lambda$ is a Kronecker delta.

Of course, the boundary term ruins it from working in the limit. That, and the lack of symmetry in mu and nu, but this should give the idea of what could work with a better choice of starting point.)

Let us begin with a Lagrangian of the form

$$\mathscr L= \frac 12 \sqrt{-g}g^{\mu\nu}\partial_\mu\phi(x)\partial_\nu\phi(x)+\mathscr L_g,$$

where $$\mathscr L_g=\frac 1{16\pi k}\sqrt{-g}R.$$ Suppose as well that there are no Killing vectors associated to the metric $g^{\mu\nu}$ except for, say, a global timelike Killing vector if it helps the argument.

Associated to $\mathscr L$ is a locally conserved set of 10 currents from the Poincare group:

$$T^{\mu\nu}=\partial^\mu\phi(x)\partial^\nu\phi(x)-g^{\mu\nu}\mathscr L$$

for each spacetime translation, and $\epsilon_{\alpha\beta}x^\alpha T^{\mu\beta}$ for each spacetime rotation.

Locally we have $$\nabla_\mu T^{\mu\nu}=0$$ so these quantities are conserved only locally.

My question is, what is the obstacle to patching these locally conserved quantities together to make a globally conserved quantity:

$$Q=\int T^{0 \nu}f_\nu \;d^3x$$

with

$$dQ/dt=0$$

where $f_\nu$ might be a gluing function connecting the momentum flowing out of one patch of infinitesimal volume and into another?

(Edit: I realize there may not be a tensor associated to this conserved quantity but even a pseudo tensor involving only the fields would be satisfying, if it exists.)

Let us begin with a Lagrangian of the form

$$\mathscr L= \frac 12 \sqrt{-g}g^{\mu\nu}\partial_\mu\phi(x)\partial_\nu\phi(x)+\mathscr L_g,$$

where $$\mathscr L_g=\frac 1{16\pi k}\sqrt{-g}R.$$ Suppose as well that there are no Killing vectors associated to the metric $g^{\mu\nu}$ except for, say, a global timelike Killing vector if it helps the argument.

Associated to $\mathscr L$ is a locally conserved set of 10 currents from the Poincare group:

$$T^{\mu\nu}=\partial^\mu\phi(x)\partial^\nu\phi(x)-g^{\mu\nu}\mathscr L$$

for each spacetime translation, and $\epsilon_{\alpha\beta}x^\alpha T^{\mu\beta}$ for each spacetime rotation.

Locally we have $$\nabla_\mu T^{\mu\nu}=0$$ so these quantities are conserved only locally.

My question is, what is the obstacle to patching these locally conserved quantities together to make a globally conserved quantity:

$$Q=\int T^{0 \nu}f_\nu \;d^3x$$

with

$$dQ/dt=0$$

where $f_\nu$ might be a gluing function connecting the momentum flowing out of one patch of infinitesimal volume and into another?

(Edit: I realize there may not be a tensor associated to this conserved quantity but even a pseudo tensor involving only the fields would be satisfying, if it exists. So for example, to get the ball rolling, we can start with an object of the form

$M^{\lambda\mu\nu}=\frac 12\int_{a^\mu}^{x^\lambda}ds T^{\mu\nu}(s)-\frac 12\int_{a^\mu}^{x^\mu}ds T^{\lambda\nu}(s)$,

and then set $t^{\mu\nu}=\partial_\lambda M^{\lambda\mu\nu}$.

$t^{\mu\nu}$ is a psuedo tensor that is conserved $\partial_\mu t^{\mu\nu}=0$ generically by the antisymmetry in $\lambda,\ \mu$, and as well

$t^{\mu\nu}=T^{\mu\nu}(x)+\frac 12\int^{x^\mu}ds\ \partial_\lambda T^{\lambda\nu}(s)+\frac 12\delta^{\mu}_\lambda T^{\lambda\nu}(x|_{x^\mu=a^\mu})$

In flat space this quantity is almost conserved up to the boundary term $\frac 12\delta^{\mu}_\lambda T^{\lambda\nu}(x|_{x^\mu=a^\mu})$ where $\delta^\mu_\lambda$ is a Kronecker delta.

Of course, the boundary term ruins it. That, and the lack of symmetry in mu and nu, but this should give the idea of what could with a better choice of starting point.)