All Questions
4,317 questions
0
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The second variational derivative of Ricci tensor with respect to metric
Is there a more efficient way to compute the second functional derivative of Ricci tensor,
\begin{equation}
\frac{\delta^2 R_{\mu \nu}(x)}{\delta g^{\alpha \beta}(y) \delta g^{\gamma \epsilon}(z)}
\...
- 85
1
vote
1
answer
28
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From material derivatives to partial derivatives in the wave equation
Consider the Cauchy momentum equation:
$$\rho \frac{d^2 \mathbf{u}}{d t^2} = \nabla \cdot \boldsymbol{\sigma} + \rho \mathbf{f}$$
where $\rho(\mathbf{x},t)$ is the density, $\mathbf{u}(\mathbf{x},t)$ ...
- 225
-1
votes
2
answers
36
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Why must the total time derivative only be a linear function of velocity? [duplicate]
I'm hung up on page 7 of Landau & Lifshitz Course on Mechanics. They claim,
$$L(v'^2) = L(v^2)+\frac{\partial L}{\partial v^2}2\textbf v\cdot \epsilon \tag{p.7}$$
The second term on the right of ...
3
votes
1
answer
118
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Covariant derivative acting on Dirac delta function
Pardon my naive computational question. In my calculations, I encounter the following expression:
\begin{equation} \label{eq1}
\frac{\delta}{\delta g^{\gamma \epsilon}(z)} \left( g_{\mu \alpha}(x) \...
- 85
1
vote
0
answers
40
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Mass Conservation in Kinetic Theory
In chapter 9 (The Boltzmann Equation) of Schwabl's 2006 text 'Statistical Mechanics', the author has the following statement of conservation of mass,
$$ \frac{\partial n}{\partial t} + \nabla \mathrm{...
1
vote
1
answer
50
views
Parallel transport of a vector on a $2d$ plane [closed]
Consider a 2d plane such that there is a curve $\gamma$ that traces a circle of radius $r=r_0$. Suppose a vector $A^\mu = (A_1, A_2)$ is attached on the circle as shown in the image below. I want to ...
- 512
-1
votes
0
answers
63
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Four gradient relation
I'm doing an exercise in QFT and I have to calculate the energy-momentum tensor for the Klein-Gordon Lagrangian and by doing it I got the following term:
$$ \frac{\partial \ \partial^{\nu}\phi}{\...
- 141
8
votes
4
answers
576
views
What kind of tensor is the electromagnetic field tensor (Faraday tensor)
I've seen the EM field tensor mostly written with 2 upper indices, $F^{\mu\nu}$. Does this imply that it's necessarily a (2,0)-tensor (that is, it has 2 contravariant components and 0 covariant ones)? ...
- 95
0
votes
1
answer
136
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Does this tensor identity hold in any kind of generality?
Assuming Minkowski spacetime, I am given an antisymmetric tensor $F^{\alpha\beta}$, and am asked to prove the following identity:
$$ F_{\mu}{}^{\alpha}{}_{,\nu}F^{\nu}{}_{\alpha} = -F_{\mu\alpha,\...
- 3,358
3
votes
0
answers
63
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How to calculate the full second-order correction to the eigenvalues in degenerate perturbation theory?
I am approaching degenerate perturbation theory from the point of view of continuum mechanics.
Consider the eigensystem:
$$C = \sum_{k=1}^{3}\lambda_{k}N_{k}N_{k}^{T}$$
where:
$C$ is a real symmetric ...
- 41
0
votes
0
answers
19
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Auxiliary Particles Green's Function Factoring - Just Tensor Algebra Calculation and Consequences?
I have a question regarding the single-particles Green's Function in Imaginary time $\tau$ for a physical fermion $c^\dagger$ expressed by the quasi-particles auxiliary fields
$$ c^\dagger_{i, \sigma} ...
- 29
3
votes
2
answers
340
views
Understanding the definition of the covariant derivative
I'm currently working my way through the book "Mathematical Methods for Physics - An Introduction to Group Theory, Topology and Geometry" and I think I have a very fundamental ...
- 61
-2
votes
1
answer
59
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Need help in understanding Tangential Acceleration [closed]
I am studying Circular motion and I am confused about tangential acceleration and tangential velocity. I am studying uniform circular motion and it says the tangential acceleration is $0$ in uniform ...
3
votes
1
answer
67
views
"Deriving" the covariant derivative
Suppose we are working in scalar QED with Lagrangian
$$\mathscr{L} = (D_\mu \phi)(D^\mu \phi)^* - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$
I now want to find the form of the covariant derivative $D_\mu$ ...
- 283
0
votes
1
answer
53
views
Derivative for the Maxwell field [closed]
I'm struggling with the following expression, which occurs in the derivation of the Maxwell Lagrangian in field theory.
$$\frac{\partial(\partial_{\mu}A^{\sigma})}{\partial(\partial^{\nu}A_{\lambda})}...
0
votes
1
answer
80
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The definition of the Lie Derivative
I am aware that an answer to an almost identical question already exist, however, I found the already existing answer not helpful (at least to my current question).
Carroll defines, in his book, the ...
0
votes
1
answer
70
views
Why are Weyl's Equations composed of only first-order derivatives?
I'm studying the Weyl's Equations from Section 1.5 of Perkins' Introduction to High Energy Physics.
The author says this:
Dirac set out to formulate a wave equation symmetric in space and time, ...
- 1,637
3
votes
0
answers
105
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Rigorous Justification for "Only Available Tensor"
In many instances, for example in particle physics, some argument of the following form is given: This rank 2 tensor we have here is a function of some vector (say $p^\mu$), so it must be some linear ...
- 497
1
vote
3
answers
89
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When transfoming an intertia tensor from one set of principle axes to another, why does it not change the tensor?
I have two inertia tensors for a cube, $I_1$ and $I_2$ where $I_1$ is the inertia tensor centered at the cube center with the axes perpendicular to the cube faces, and $I_2$ is also centered at the ...
- 11
9
votes
4
answers
4k
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Is it ever possible that the object is moving with a velocity such that its rate of change of speed is not constant but acceleration is constant?
Is it ever possible that the object is moving with a velocity such that its rate of change of speed is not constant, but rate of change of velocity is constant?
Like speed is only the magnitude, so ...
1
vote
1
answer
50
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Grassmann Numbers, anticommutation and derivative rules
If $\psi(t)$ is a complex Grassmann number and $\psi^*(t)$ is its complex conjugated. The following is true:
$$\frac{\partial (\psi^*\psi)}{\partial \psi}=-\psi^*\frac{\partial \psi}{\partial \psi}=-\...
- 1,628
-2
votes
0
answers
70
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Use of $dv/ds$ in defining acceleration [duplicate]
We can write acceleration as either
$dv/dt$ or $v dv/ds$.
And surprisingly the work-energy theorem arrives from the second definition.
I feel it would be fundamentally understanding towards work ...
-1
votes
0
answers
17
views
How to prove that a Lie algebra-valued differential form is exact for the covariant derivative [migrated]
Given a differential $p$-form $\omega^A$ over a smooth manifold with values on some Lie algebra, I wanted to know how could one prove that it can be written as an exact form for the exterior covariant ...
- 381
0
votes
0
answers
59
views
What's the difference between $dx$ and $\delta x$? [duplicate]
In the process of defining crystal momentum $\hbar k$, I found these formulas below.
By the definition of group velocity,
$$v_g=\frac{d\omega_{nk}}{dk}=\frac{1}{\hbar}\frac{dE_{nk}}{dk}$$
Also if an ...
- 51
0
votes
1
answer
66
views
Is Stress a Derivative?
On page 289 of the text "Fundamentals of Fluid Mechanics" by Munson et al., the authors give the following definition of the normal stress acting on the surface of a fluid element:
At any ...
0
votes
0
answers
80
views
Second Derivative of Antisymmetric Tensor Vanishes?
Consider the rank two antisymmetric tensor field defined over some manifold with metric $g$ (named $F$ because I did this work for E&M):
$$F^{\alpha \beta}=-F^{\beta \alpha}$$
Now consider acting ...
2
votes
3
answers
445
views
Rank-2 tensor multiplication discrepancy
Given any tensor $A \,\text{and}\, B$. (Not necessarily symmetric or anti-symmetric)
$$A^{\mu}_{\,\,\,\nu}B_{\mu}^{\,\,\,\rho} = C_{\nu}^{\,\,\rho} \,\,\, \text{or} \,\,\, C^{\rho}_{\,\,\,\nu}\,\,\,?\...
- 95
3
votes
1
answer
480
views
Second derivative of unit vector
We know that the second derivative of unit vector (the vector from a point toward the source) is proportional to the Electric field caused by the source in a particular point.
If we imagine that our ...
- 63
3
votes
1
answer
114
views
Relationship between covariant derivative and metric tensor
In general relativity, the covariant derivative of the coordinate vector is a tensor, equal to $$x^{\mu}_{:\rho} = x^{\mu}_{,\rho} + \Gamma^{\mu}_{\rho\nu}x^{\nu},$$ is it meaningful to equate this ...
- 562
1
vote
0
answers
76
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Confusion on coordinate transformation matrix derivatives and Christoffel symbol transformation
Consider the coordinate transformation from $x^{\mu} \to x^{\bar{\mu}}$ given by the transformation matrix $\Lambda^{\bar{\mu}}_{\mu}$, and $\Lambda_{\bar{\mu}}^{\mu}$ for the inverse transformation.
...
- 41
3
votes
3
answers
736
views
Einstein's index notation for symmetric tensors
Writing the product
$$
\mathbf{T} = \mathbf{A}^\text{T}\boldsymbol{\eta}\mathbf{A}\tag1
$$
in Einstein index notation I get:
$$
T_{\mu\nu} = (A^\text{T})^{\alpha}_{\,\,\mu}\eta_{\alpha\beta}A^{\beta}_{...
- 57
4
votes
2
answers
243
views
Leibniz rule and Nakahara's definition for functional derivatives with respect to Grassmann variables
In Nakahara's book "Geometry, Topology and Physics" in section 1.5.7 (I'm reading the second edition) he defines the functional derivative with respect to Grassmann variables. He does so in ...
- 43
3
votes
2
answers
79
views
When does the rank of a tensor reduce when we integrate it over a 3D foliation of spacetime?
In physics, we have the idea that the integral over $d^3x$ of the 0th component of energy-momentum density $T^{\mu \nu}$:
$P^{\mu}=\int d^3x T^{\mu 0}$
transforms like a Lorentz four-vector.
Similarly,...
- 6,666
1
vote
0
answers
62
views
A trick for derivatives of thermodynamic quantities [closed]
Starting from
$$dU=TdS-PdV$$
We can write, for instance $U(T,V)$ and $S(T,V)$ to obtain:
$$\left(\frac{\partial U}{\partial T}\right)_VdT+\left(\frac{\partial U}{\partial V}\right)_T dV=T\left(\frac{\...
- 225
3
votes
0
answers
66
views
Globally hyperbolic spacetimes and (1+3) decomposition
I am having trouble in understanding the relation between globally hyperbolic manifolds and the (1+3)-decomposition used in general relativity. Let me start with the following two preliminaries:
Let $...
- 884
0
votes
1
answer
55
views
How to derive the stress tensor in quantum mechanics?
Nielsen in his article Phys. Rev. B 35, 9308 (1987) derived the stress via a stretched particle coordinate $\mathbf{r}_{i\alpha}\rightarrow \mathbf{r}_{i\alpha}+\sum_\beta \epsilon_{\alpha\beta}\...
- 31
0
votes
0
answers
36
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Is this mathematically correct that gradient of deformation gradient is equal to deformation gradient?
The deformation matrix is defined as follows, where $x$ is the current location and $X$ is the reference location. It shows the relationship between current $x$s with regard to original $X$s,
$$F = \...
- 21
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0
answers
21
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Conflicting Solutions for Calculating Apparent Speed of Jogger's Image in Convex Mirror
I’m facing a challenge with a physics problem due to conflicting solutions across different sources, and I'd appreciate some clarification.
Problem Statement:
Suppose, while sitting in a parked car, ...
1
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0
answers
54
views
Understanding this formula in Thorne (1980)
I am studying this paper by Thorne:
Multipole expansions of gravitational radiation,
link to pdf
I am trying to figure out equation (2.12) pag. 11 which links the Symmetric Trace Free (STF) tensor ...
- 1,187
0
votes
0
answers
72
views
Are line elements vectors or covectors?
In continuum mechanics literature I regularly read statements such that the deformation gradient $\mathbf{F}$ of the flow map $\boldsymbol\varphi$ transforms line elements $\mathrm{d}\mathbf{X}$ from ...
- 71
0
votes
1
answer
44
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Rotational averages of tensors
I was recently thinking about taking a rotational average of a tensor - i.e., take all the possible rotations of the tensor and average them. A physical example of this could be getting isotropic ...
- 884
2
votes
1
answer
59
views
One-form transformation law
Suppose $V^\mu$ is a vector. Let $V_\mu$ be the corresponding one-form obtained by contracting $V$ with the Minkowski metric $\eta_{\mu \nu}$. Then
\begin{equation}
V_\mu = \eta_{\mu \nu}V^\nu.
\...
2
votes
1
answer
80
views
Is the quadrupole tensor a $(2,0)$ tensor?
In classical electromagnetism, the quadrupole moment is usually written as
$$Q_{ij}=\int d^3r \rho(\vec{r})(3r_ir_j-r^2\delta_{ij}). \tag{1}$$
However, this represents the quadrupole tensor as a ...
- 3,295
3
votes
1
answer
94
views
What happens to $g^{\alpha\beta}_{,\sigma}=-g^{\alpha\mu}g^{\beta\nu}g_{\mu\nu,\sigma}$ when $g_{\mu\nu}\rightarrow \eta_{\mu\nu}$ (weak field limit)?
The equation
$$g^{\alpha\mu}_{\,\,\,\, ,\sigma}\,g_{\mu\nu} + g^{\alpha\mu}\,g_{\mu\nu,\sigma} = (g^{\alpha\mu}g_{\mu\nu})_{,\sigma} = \delta^\alpha_{\nu,\sigma} = 0 $$
gives the useful relation
$$g^{\...
- 467
0
votes
3
answers
105
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What is the physical significance of the electromagnetic tensor $F^{\mu\nu}$ having its indices on top or bottom? [duplicate]
I understand we need to put indices on a four vector on the top or bottom like $A^{\mu}$ or $A_{\mu}$ because of the invariant Lorentz product having its later three terms negative (for $+ - - -$ ...
0
votes
2
answers
64
views
Why the EOM from the variation of the metric coincide with the EOM from the variation of an ansatz for the metric?
Suppose we have a pure gravitational action $S[g_{\mu\nu}]$ and we obtain, via the usual variation procedure, its associated Equation Of Motion (EOM) for any $g_{\mu\nu}$. After that, we can give an ...
- 77
26
votes
21
answers
5k
views
What happens when a car starts moving? The last moment the car is at rest versus the first moment the car moves
Imagine a car that's at rest and then it starts moving. Consider these two moments:
The last moment the car is at rest.
The first moment the car moves.
The question is: what happens between these 2 ...
- 371
0
votes
0
answers
38
views
Four-divergence of a vector [duplicate]
The covariant derivatives of a four-vector is
$$
\nabla_{\nu}U_{\mu} = \partial_{\nu}U_{\mu} - \Gamma^{\lambda}_{\mu\nu}U_{\lambda}
$$
It has the following identity:
$$
\nabla_{\mu}U^{\mu} = \frac{\...
- 49
4
votes
3
answers
381
views
Relationship between tensor product in Lie algebra and in quantum mechanics
If we treat dynamic operators like $P$ in quantum mechanics as generators in Lie algebra,and the square of operators like $P^2$ as elements in the universal enveloping algebra.
Then in this wiki https:...
- 184
0
votes
0
answers
54
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Magnetic dipole field from vector potential
The magnetic dipole field:
$$\textbf{B}=\frac{\mu_0I}{4\pi}\frac{3(\textbf{m}\cdot\hat{\textbf{x}})\hat{\textbf{x}}-\textbf{m}}{r^3}$$
can be calculated from the vector potential $\textbf{A}$ and ...
- 172