6
votes
Physical significance of metric compatibility
There is a formulation of GR where metric compatibility follows from the equations of motion. If you think of the Einstein-Hilbert action as being a functional of both the connection and the metric (...
- 33.3k
5
votes
Accepted
Ricci Identity with Torsion Proof
Hint: You can, in a first step, expand the outer derivative (write $D_\nu Z^\sigma=A_\mu^\sigma$ if you wish). You will get a partial derivative acting on $A_\mu^\sigma$ and two terms with ...
- 3,574
5
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Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $
It's not unreasonable to be confused about this. Let's say you have a function $f$ of one variable and $f'$ is its derivative. Then we have
$$\frac{d}{dx} f(c-x) = \color{red}{-}f'(c-x)$$
via the ...
- 50.8k
4
votes
Ricci Identity with Torsion Proof
Remember that $\nabla_{\nu} Z^{\sigma}$ is a type $(1,1)$ tensor, so $\nabla_{[\mu} \nabla_{\nu]} Z^{\sigma}$ will spit out terms like $\Gamma_{[\mu \nu]}^{\lambda} \nabla_{\lambda} Z^{\sigma}$.
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3
votes
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Non-parallel light diffraction
Yes it does. A common version on the double slit experiment is, in fact, the one you describe. You can, for instance, install a narrow hole with the primary light source behind it. This hole will act ...
- 613
3
votes
What are the spaces in which quantum fields belong and how does that affect the hermitian conjugate of $\partial_{\mu}$?
First, let's consider the derivative $\partial$ as an operator on the Hilbert space $\mathcal{H} = L^2(\mathbb{R}^n)$ of square-integrable functions on $\mathbb{R}^n$ (the space typically encountered ...
- 361
3
votes
Ricci Identity with Torsion Proof
We have
$$\nabla_\mu \nabla_\nu Z^\sigma=\partial_\mu(\nabla_\nu Z^\sigma)+\Gamma_{\lambda\mu}^\sigma \nabla_\nu Z^\lambda - \Gamma_{\nu\mu}^\rho \nabla_\rho Z^\sigma$$
as $\nabla_\mu$ is acting on ...
- 360
3
votes
Accepted
Rigorous treatment for continuous mass systems
In discrete case we sum over particles, in continuous case we integrate over some space, physical or coordinate.
When calculating e.g. coordinates of center of mass of some continuous body, summation ...
- 27.8k
3
votes
Accepted
Notation and Terminology Questions from Schwartz' QFT Book
Yes. OP is right. The LHS of eq. (3.30) is supposed to be a partial derivative.
Yes. OP is right. The RHS of eq. (3.31) is supposed to be a total derivative.
See also this related Phys.SE post.
- 167k
2
votes
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Four-vector differentiation (E-M Euler-Lagrange eq.)
As mentioned in the OP's link Schwartz (in his QFT book) doesn't keep track of the index placement on tensor objects that might obscure the structure a little.
However, ignoring the first derivative $\...
- 407
2
votes
Physical significance of metric compatibility
Metric compatibility follows from the Einstein equivalence principle, which asserts that the laws of special relativity hold for local non-gravitational phenomena in a freely falling laboratory. Here, ...
- 3,195
1
vote
Accepted
Generalizing Fermi-Walker Derivative/Transport to General Vector Bundles
Rather than viewing the Fermi-Walker derivative as a specific differential operator (which is what is causing the issues with generalization and the dependence on the curves $\gamma_1,\gamma_2$), one ...
- 2,304
1
vote
Accepted
Srednicki 11.3 part e) Finding the maximum energy for the electron
This looks like a perfect storm of miscommunication. You appear overwhelmed by tangential points and to be construing problems that are not there.
First, independently of the rest/bulk of your ...
- 47.2k
1
vote
Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $
The point is that $E$ is fixed, so that if you raise $E_1$ by some amount $\delta x$ you have to lower $E_2$ by the exact same amount $\delta x$. This means that $E_1$ and $E_2$ can not be varied ...
- 7,006
1
vote
Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $
In my opinion the notation is very bad. $S_2$ is a function of a single variable (at least in this context here) and it simply does not make sense to compute partial derivatives with respect to two ...
- 3,482
1
vote
Accepted
Velocities - Equation 1.46 of Goldstein 3rd edition
He is using the standard chain rule for partial differentiation from calculus. The partial derivative and the total derivative are not the same. See any calculus text such as one by Kaplan, Thomas, or ...
- 5,825
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