# Tag Info

1

From my readings; the key to conservation of momentum appears to be based on defining a closed system to see if any mass crosses the boundaries of the system.

2

You cite the example of a (Newtonian) gravitational potential which, naively, seems to depend on the position of the bodies under consideration. However, a little more contemplation is warranted here. Consider the following: When we ask a basic question about the gravitational potential energy of some body relative to Earth, do we have to specify where ...

1

Gauss-Ostrogradsky theorem does not actually "care" for a "type" of space or functions that you've got, so it has the same form in curved space and in flat space: $$\int_D d^4x \frac{\partial}{\partial x^\mu} Anything = \oint_{\partial D} d\sigma_\mu Anything$$ where $d^4 x$ and $d\sigma_\mu$ are constructed from differentials of coordinates the same ...

3

Disclaimer: Let us here avoid the discussion of how to assign a stress-energy-momentum (SEM) pseudo-tensor $t^{\mu\nu}$ to the gravitational field. The word pseudo here refers to the fact that $t^{\mu\nu}$ is not a tensor wrt. general coordinate transformations; only a rigid subgroup thereof. In other words, the pseudo-tensor ...

4

This is how we taylor expand the determinant to first order \begin{align} \mathcal{J}&= \text{det}\left(\frac{\partial x'^j}{\partial x^i} \right) \\ &= \text{det}\left(\delta _i ^j + \partial_i\delta x^j\right) \\ &= \text{exp}\left( \text{tr log }\left(\delta _i ^j + \partial_i\delta x^j\right)\right) \\ &= \text{exp}\left( \text{tr ...

1

Ali moh's is a wonderfully full and descriptive answer, but unfortunately you are not looking for a purely mathematical answer in which case I will try to give the kind of answer you are looking for. 1) The reasons associated with the variations being described in terms of these sums have to do with perturbation theory (which goes beyond the scope of this ...

2

First of all $\delta$ and $\frac{\partial}{\partial x}$ don't commute because $$\delta \left(\frac{\partial}{\partial x}\phi\right) = \delta\left( \frac{\partial}{\partial x}\right)\phi + \frac{\partial}{\partial x}\left(\delta\phi\right)$$ Second we divide $\delta\phi_I = \bar{\delta}\phi_I + X^k_n \delta \omega_n \partial_k \phi_I$ because the total ...

1

Before going to field theory, it seems instructive to first ask the same questions in point mechanics: Can the Lagrangian $L(q,v,t)$ depend on time explicitly? Yes. The Lagrangian $L(q,v,t)$ can depend explicitly on time. E.g. there could be external sources. On the other hand, if the Lagrangian does not depend on time explicitly, then the ...

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