2
votes
1answer
53 views

First variation of the action in relativistic notation - Landau & Lifshitz “Classical theory of fields”

In Landau & Lifshitz's book, Classical theory of fields, the action for a free particle is defined as: $$\tag{8.1} S= \int ^b _a {-mc \ \text d s}=0,$$ where $$\text d s=c\,\text d ...
0
votes
1answer
65 views

Rotational symmetry in integration

Can someone please tell me why $$4\int d^4x \, x^\mu x^\nu ~=~\int d^4x \, g^{\mu\nu}x^2 $$ by some rotational symmetry argument?
3
votes
0answers
50 views

The question about MTW 4-momentum integral expression and lorentz nature

In section 5.8 of Misner, Thorne, and Wheeler's "Gravitation" there is a proof that 4-momentum determined as $$ \tag 1 p^{\mu} = \int T^{\mu 0}\,\mathrm{d}^{3}\mathbf r , \quad \partial^{\mu}T_{\mu ...
1
vote
3answers
935 views

Integration by parts to derive relativistic kinetic energy

I have come across a weird integration during derivation of relativistic kinetic energy. Our professor states that i can get RHS out of LHS using integration by parts: $$ \int\limits_0^x \! ...
3
votes
0answers
191 views

Should the Jacobian be negative in $\mathrm{d}^4 x$?

In page 24 of Srednicki's QFT textbook, he says that $\mathrm{d}^4x$ is a Lorentz scalar. I understand that the determinant of a Lorentz matrix is always $\pm 1$. So in an improper Lorentz ...