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1answer
58 views

General Relativity - Four Velocity Derivative Question

I am trying to get my head around a small point used in a book I am reading about General Relativity. The book states that because $u_au^a = c^2$ it follows that $u_a \nabla_b u^a = 0 $ The first ...
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0answers
129 views

Is it correct to sum over either index of the metric the same way?

I don't know if the following is correct, i want to compute the following derivative $$\frac{\partial }{\partial (\partial_{\mu}A_{\nu})}\left(\partial^{\alpha}A^{\beta}\partial_{\alpha}A_{\beta} ...
4
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0answers
296 views

Why is the $\theta$ term of QCD violating charge and parity (CP) symmetries?

From the non-trivial nature of the QCD vacuum, the Lagrangian is augmented with a term like \begin{equation} \theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu} \end{equation} where $ ...
3
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0answers
114 views

Question about derivation of tensor in Di Francesco's CFT

This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two ...
3
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0answers
131 views

Can the two electromagnetic field tensors be combined into a more general tensor?

Given the electromagnetic field tensor $$\begin{align} F_{\mu\nu} = \begin{pmatrix} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & B_{z} & -B_{y} \\ E_{y} & -B_{z} & 0 ...
3
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0answers
384 views

I lost a factor of two in the electromagnetic field tensor

I apologize for this simple question, but I lost a factor of 2 and can't find it anymore, so now I'm looking on the internet, perhaps one of you has some information about its whereabouts. :-) ...
2
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0answers
87 views

Calculation of Einstein Equation

I have a 3d system with Lagrangian $$e_3^{-1} L_3 = -\frac{1}{2} R_3 + \delta_{ab} \partial_\rho q^a \partial^\rho q^b + \frac{1}{2H} V(q)$$ From this I want to calculate the Einstein equation by ...
2
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0answers
92 views

Questions about closed forms and cycles

I read the section closed forms and cycles in Arnold's Mathematical Methods of Classical Mechanics (page 196-200), but the problems in this section is too difficult to solve in the way following the ...
2
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0answers
101 views

Stress calculations in a perforated paper

You have a sheet of paper (torn out of a good quality foolscap notebook) as shown above, and you start pulling it apart with both your hands (forces indicating by the blue arrows). Its difficult to ...
2
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0answers
138 views

Lecture Notes confusion: Constructing the Einstein Equation

This question is on the construction of the Einstein Field Equation. In my notes, it is said that The most general form of the Ricci tensor $R_{ab}$ is $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ ...
1
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0answers
54 views

How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
1
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0answers
29 views

Is energy-momentum of curvature a boundary/holographic density?

Since the beginnings of General Relativity, we have had this awkward, unholy separation of the universe in marble versus wood. divergence of the stress-energy momentum holds at all points of ...
1
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0answers
118 views

Energy-momentum tensor

I need to show that: \begin{align} \mathcal h_i^a \, T_{ab} \, h_i^b=(\nabla_i \phi)^2-\frac{h_{ii}}{2}[\dot{\phi}^2-(\nabla \phi)^2-m^2 \phi^2] \end{align} where i) $T_{ab}=\nabla_a \phi ...
1
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0answers
40 views

Derivative of a constant tensor field along a path

I understand that the time derivative of a tensor filed $\mathbf T$ along a curve $\gamma\left(t\right)$ can be shown to be, $$ \frac { d\mathbf{T}}{ dt } = \left(\frac { dT^{ i } }{ dt } + V^{ k ...
1
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0answers
41 views

Just a contraction of indices

I came across a contraction which is not giving the desired result. This is a toy problem in how to get a supergravity theory in low energy limit of a superstring theory using the vanishing of beta ...
1
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0answers
116 views

Variation of the purely covariant Riemann tensor

I need to find the variation of the purely covariant Riemann tensor with respect to the metric $g^{\mu \nu}$, i.e. $\delta R_{\rho \sigma \mu \nu}$. I know that, $R_{\rho \sigma \mu \nu} = g_{\rho ...
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0answers
105 views

Index Notation Double Curl

My question is about Einstein notation. It does not matter the specifics of this example (the del operator could be another random vector), I just want to know if my assumption about notation is ...
1
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0answers
79 views

What is $T_{\mu\nu}T_{\mu\nu}$ for the electromagnetic stress-energy tensor?

Given the electromagnetic stress-energy tensor components \begin{align} T_{\mu\nu} = \begin{pmatrix} u_{00} && s_{0 \nu} \\ s_{\mu 0} && \sigma_{\mu\nu} \end{pmatrix}_{\mu\nu} ...
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0answers
43 views

How to calculate the minimum number of extrinsic dimensions of a metric tensor?

The Question How does one calculate the minimum number of dimensions of an extrinsic space that can be used to define the metric tensor \begin{align} g_{mn} = \dfrac{\partial y^k}{\partial x^m} ...
1
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0answers
128 views

How to integrate twice of this viscous term?

I am reading a paper, and I do not understand why the author said the following term when integrated twice will become, $\int\limits_\Omega {{\rm{d}}\Omega {{\bf{\psi }}^{\bf{u}}}\cdot\nabla ...
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0answers
38 views

Finding inverse tensor operator

For example I have such tensor operator: $$ O^{\mu \nu \alpha \beta} = (a^2+m^2)(\eta^{\mu\nu} \eta^{\alpha\beta} + \eta^{\mu\alpha} \eta^{\nu\beta}) + a^\mu a^\nu \eta^{\alpha\beta} + a^\mu ...
0
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0answers
49 views

metric determinant and its partial and covariant derivative

question : $\nabla_a \nabla_b \sqrt{g} \phi =\partial_a \sqrt{g} \partial_b \phi$ is true ? because $\nabla_a \sqrt{g}=0$ so we can write $\sqrt{g} \nabla_a \nabla_b \phi$ , but because metric ...
0
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0answers
17 views

I wasn't able to find a good resource for Bipartite state and Bell's theorem

Our professor used tensor product to explain bipartite operator and states and then he used the new operator and state to explain Bell theorem. I wasn't able to find a good resource or reference for ...
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0answers
61 views

Invariant Form of The Material Derivative

Why is the RHS of the following equation invariant to coordinate transformation and the LHS is not? And is there a way to show the equivalency between the LFS and RHS? \begin{align} \vec{V} \cdot ...
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0answers
65 views

Why doesn't this proof change indices?

In this pdf, in the second line of the proof, $\sigma$ was plugged in where it appears as $$\frac{\partial x^\sigma}{\partial y^{\rho'}}$$ Meanwhile in converting the coordinates of $g^{\mu'\rho'}$, ...
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0answers
68 views

Jacobian for Kronecker delta

I was revising on a bit of tensor calculus, when I stumbled upon this: $$\delta^i_j = \frac{\partial y^i}{\partial x^\alpha} \frac{\partial x^\alpha}{\partial y^j}$$ And the next statement reads, ...
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0answers
85 views

Einstein frame vs. Matter frame

What is the difference between Einstein frame and Matter frame in General Relativity? -A brief comment on each could be useful too. These two frames were used in this manuscript ...
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0answers
72 views

How should the implicit sum $C_{ijkl}u_{i,j}u_{k,l}$ be interpreted?

$C$ is a 3x3x3x3 tensor. How should the expression $C_{ijkl}u_{i,j}u_{k,l}$ be interpreted? This is my guess: $$ \sum_{i=1}^3\sum_{j=1}^3 \sum_{k=1}^3\sum_{l=1}^3 C_{ijkl}u_{i,j}u_{k,l} $$
0
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0answers
73 views

Angular Momentum with Upper Index

I am asked to show $[L^2,L_i] = 0 $, but with the definition : $L^2 \equiv L_i L^i$ I tried this: $[L_i L^i,L_i] = L_i [L^i,L_i] + [L_i,L_i]L^i$ We know that : $[L_i,L_i]$ = 0 , so we have, $[L_i ...
0
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0answers
477 views

covarient derivative of electromagnetic field tensor

I'm trying to prove the energy momentum tensor in curved spacetime for Electromagnetic field is Divergence-less directly(Without using general lie derivative method which can prove any energy momentum ...
0
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0answers
84 views

How do I write the energy of a constant, uniform 2D charge distribution?

Let's consider a 2D electromagnetic field defined in a square domain $[0,\Lambda]^2$, with periodic boundary conditions, with a constant charge distribution, uniform all over the aforementioned ...