Questions tagged [stress-energy-momentum-tensor]
A symmetric rank-2 tensor in relativity, which expresses the flux of energy-momentum along timelike and spacelike axes. Also known as the stress tensor or the energy-momentum tensor. In the Einstein field equations, it is the source of gravitational fields.
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Time dependence of momentum operator in QFT
In QFT in flat spacetime, we want to have a representation of the Poincare algebra on the Hilbert space $\mathscr H$ of states. This is due to the Wightman axioms. So there should exist operators $M_{\...
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Does a change in the trace of the stress-energy tensor when a particle passes through an event horizon violate energy conditions?
When a particle crosses the event horizon of a black hole, the trace of its stress-energy tensor undergoes a change in sign, prompting questions about potential violations of the strong energy ...
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What are the strictly-required energy conditions for valid spacetime?
In learning general relativity, I am getting confused about which energy conditions (weak, strong, null, dominant) must actually be fulfilled for a spacetime to be considered "valid" in the ...
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Calculating the energy momentum tensor of $bc$ and $\beta\gamma$ CFT
I'm trying to calculate the energy momentum tensor of the $bc$ and $\beta\gamma$ CFT's, using a simple method of calculating the canonical expression and adding total derivatives (rather than the ...
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Why don't non-local theories typically have any energy-momentum tensor?
This great answer by @AccidentalFourierTransform says that for energy momentum tensors we need locality and Lorentz invariance. The rest of the answer focuses on metric dependence of partition ...
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QFT schwartz chapter 3b. total derivative
I am trying to improve my knowledge of QFT and was going through Schwartz's QFT. In question. 3.2 you are asked to evaluate how $$Q=\int \mathcal{T}_{00} d^3 x$$ changes when you add a total ...
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Polarization of gravitational waves - Dirac's rotation operator
I refer to the page extracts below. I think my question is fully self-contained, but for background:
Dirac is dealing with the weak-gravity case and gravitational plane waves moving in the $l_\sigma$ ...
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Can gravitational energy be localized in the case of plane waves?
Reading Dirac's "General Theory of Relativity", Chap. $33$ "Gravitational waves". He shows that in a weak gravitational field ($g_{\mu\nu}$ approximately constant), using harmonic ...
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The electromagnetic Lagrangian $L=(8\pi)^{-1}(E^2-B^2)$ vanishes for an EM plane wave. Why doesn't the energy-momentum tensor also vanish?
The electromagnetic field Lagrangian is $$L=(8\pi)^{-1}(E^2-B^2)\,\, ,$$ which vanishes everywhere for a plane wave since the electric and magnetic fields have the same magnitude.
$\quad\quad$ By the ...
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Hawking and Ellis Lemma 4.3.1 Proof
I have a few questions about Hawking and Ellis' proof of this lemma (pages 92-93):
Write the $(2, 0)$ stress-energy tensor in coordinates as
$\mathbf{T} = T^{ab} \partial_a \otimes \partial_b$ and ...
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Is Poynting vector $E \times B$ and energy density $(E^2+B^2)/2$, a four vector?
As the title suggests, does the 00th and 0i the components of the electromagnetic energy-momentum tensor viz. Poynting vector and energy density form a four-vector?
Answers with both theoretical and ...
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Energy of the gravitational field within a sphere of radius $R$ in the Schwarzschild metric
The Landau-Lifshitz energy-momentum pseudotensor $t^{μν}$ is defined by
$$16πt^{μν}
= -2G^{μν} - g^{-1} \left[ -g \left( g^{μν}g^{αβ} - g^{μα}g^{νβ} \right) \right]_{,αβ}$$
where $g=\text{det}[g^{μν}]...
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Analog between Electromagnetism and Gravity
Feynman makes an analogy between EM field and gravity field in his Feynman's Lectures on Gravitation. The vector field representing EM potential would couple to the current source(vector) in the ...
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Does the isotropy definition of a perfect fluid imply no heat conduction?
Weinberg defines a perfect fluid (Chapter 2, Section 10) as one where each fluid element appears isotropic in a reference frame moving with that element.
From the definition of the stress-energy ...
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Energy-momentum tensor and equation of motion in Einstein-Dilaton theory [closed]
I am following this paper (see eq. 19-22) and trying to derive the equation of corresponding to Einstein-Dilaton gravity (ignoring the Maxwell part for now)
\begin{align}
S_{\text{E-D}} = \int d^4 ...
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How can the stress-energy tensor $T^{μν}$ be unique when the Lagrangian $L$ is not?
A relation (or definition) between the stress-energy tensor and the Lagrangian in GR is routinely seen:
$$T^{μν} = -2 \frac{∂L}{∂g_{μν}} - g^{μν} L \quad\quad(*)$$
(or some variation of this, ...
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Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?
I'm working through Chap. $30$ of Dirac's "GTR" where he develops the "comprehensive action principle". He makes a very slick and mathematically elegant argument to show that the ...
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Zero stress in $z$ components for thin surfaces
We can write the stress tensor as:
\begin{equation}
T= \left [
\begin{array}{ccc}
\sigma_r & \tau_{r\theta} & \tau_{rz} \\
\tau_{\theta r} & \sigma_\theta & \tau_{\...
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How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?
I want to experiment with this relation (from Dirac's "General Theory of Relativity"):
$$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$
using the electromagnetic Lagrangian $L = -(...
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If a slice of a 4 dimensional metric violates an energy condition, does the 4 dim metric violate it aswell?
I 'm currently studying analogue gravity see this paper for a review. Here a 2+1 dimensional metric is derived:
$$
ds^2 = -dt^2 + (dr - \frac{A}{r} dt)^2 + (r d\theta - \frac{B}{r} dt)^2
$$
Now it ...
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Are the Virasoro constraints generated from the Polyakov action first-class constraints in Dirac's sense?
The Polyakov action for strings reads
$$
S[X] = -\frac{T}{2} \int d^2\sigma\, \sqrt{h}h^{\alpha\beta} \partial_\alpha X^\mu \partial_\beta X_\mu,
$$
from which the Virasoro constraints follow:
$$
T_{\...
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Recovering the classical Navier-Stokes equation from Landau's relativistic tensor [closed]
I'm trying to get to the classical compressible Navier-Stokes equation from the relativistic tensors in the Landau/Eckart frame. I understand these have their own problems about entropy.
In several ...
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Field equations for systems violating local conservation rule
Is there a modified gravity theory which only allows for some weaker form of local conservation rule? Like in Einstein's gravity, equating $G_{\mu\nu}$ with $T_{\mu\nu}$ naturally leads to ...
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Electromagnetic interaction in the stress-energy momentum tensor
In Phys. Rev. D 4, 2185, the author uses the following stress-energy tensor for a charged spherically-symmetric fluid:
$$ T^{\mu \nu} := (\delta + P) u^{\mu} u^{\nu} + P g^{\mu \nu} + \frac{\pi}{4} [ ...
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How is it that energy of matter yields gravity if the amount of energy in a system is frame dependent while the force caused by gravity is not?
I've been told that the gravitational field arises due to the energy density terms in the stress-energy tensor of matter and therefore that all energy of matter exerts a gravitational field effect, ...
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Vacuum solutions in presence of mass?
Here is the page I will be referencing: Vacuum solution (general relativity) - Wikipedia
My point is: if $T_{\mu\nu}=0$ implies that there is no mass, how can Schwarzschild vacuum be a solution, if ...
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About stress-energy tensor
I am reading the classical mechanics by Goldstein. I have a trouble to understand the definition of stress-energy tensor, in particular, the index of the tensor.
The definition I learned is $$T_\mu^\...
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How do I compute the stress-energy tensor for a simple system of $N$ point particles?
I haven't been able to find a simple self-contained definition of the stress-energy tensor as used in the Einstein field equations.
Suppose I have $N$ classical (not quantum) point particles with ...
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Stationary dust solution in general relativity
I tried to calculate a stationary dust solution in general relativity where the energy-momentum tensor is $T^{\mu\nu} = \rho c^2 \delta^\mu_0 \delta^\nu_0$. (related question) The question is that I ...
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On the existence of Gravitational energy in GR [duplicate]
I was reading this paper that puts forward the argument that Gravitational energy in GR is unnecessary and doesn't exist and that got me wondering if this is a fringe theory or what exactly is the ...
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Theorem in mechanics relating energy flow and momentum
In Feynman's Lecture 27 on Vol. II it is written that
There is an important theorem in mechanics which is this: whenever there is a flow of energy in any circumstance at all (field energy or any ...
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Reason for stress and strain
Do you think that the stress is due to strain or vice versa?
I have this doubt because of two of the following scenarios:
Consider the case of the rigidly fixed bar. It is now heated (say be some ...
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Is electromagnetic pressure (stress energy tensor) positive or negative in the direction of the electric field (assuming $B=0$)?
Suppose we have an electric field pointing along the $x$-axis, and $B = 0$. The diagonal spatial components of the stress-energy tensor will all have the same magnitude, but the $y$ and $z$ components ...
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Turning a Lagrangian contains superscript and subscript indices into energy
I'm recently reading the book "Solitons and Instantons" written by R. RAJARAMAN. However, for lacking of ability, I couldn't figure out how to derivate the static solution for energy with ...
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Finding stress associated to a free energy
I am trying to find the stress tensor associated to the following free energy
$F = \int dV [ a |\nabla L|^2 + b |\nabla^2 L|^2]$
Where L is the heigh of the fluid interface, and is only a function of $...
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Derivation of transformation law for the Hilbert Stress-energy tensor [duplicate]
The Hilbert stress-energy tensor is defined as
$$T_{\mu\nu}=-2 \frac{1}{\sqrt{g}}\frac{\delta S_M}{\delta g^{\mu\nu}}.$$
Given the name one expects that it transform as a tensor, but how to prove this ...
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How does the conformal scaling behavior of the stress-energy tensor depend on the spacetime dimension?
I'm new to Weyl transformations and am struggling to find online the answer to what should be a simple question. Consider an $n=D+1$ dimensional spacetime with metric $g_{ab}$ and some stress-energy ...
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Regarding Derivation of Einstein Field Equations
In most sources I come across that try to justify the Einstein Field Equations outside the context of Einstein-Hilbert action, the argument goes mostly as follows: In analogy with the Poisson equation ...
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Why for the electromagnetic force we need "Maxwell's Stress Tensor" to be a tensor to find flux through a surface?
If we are computing the flux of a fluid through a surface, then we can simply integrate the dot product of the fluid flow through the normal component of an element of surface area... No tensor is ...
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Stress-energy tensor of a $2\text{d}$ conformal field theory [closed]
Does stress-energy tensor of a $2\text{d}$ conformal field theory split into holomorphic and anti-holomorphic parts as follows?
In a conformal field theory, stress-energy tensor $$T_{\mu\nu} = \frac{1}...
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How do gravitational waves carry energy when gravitational energy cannot be localised?
I have a very naive question, actually someone asked it and I can't answer. It simply asks that if gravitational energy cannot be localised (we cannot write a pure gravitational energy momentum tensor)...
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Generalizing the von Mises Criterion to Complex Stress Tensors
I was deriving the von Mises maximum distortion energy criterion:
$${\displaystyle \sigma _{\text{v}}={\sqrt {{\frac {1}{2}}\left[(\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{...
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The meaning of stress tensor conservation in general relativity [duplicate]
In general relativity one has the Hilbert stress-energy tensor defined as
$$T^{\rm matter}_{ab} = -\frac{2}{\sqrt{-g}}\frac{\delta S_{\rm matter}}{\delta g^{ab}}~,$$
which is covariantly conserved i.e ...
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Terms in the Israel Junction Conditions
I'm confused about the Israel Junction Conditions. I've seen them written several different ways so far, but here I'll use: $$K^-_{ij}-K^+_{ij}=8\pi(S_{ij}-\frac{1}{2}g_{ij}S).$$
My understanding is ...
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Stress-energy tensor of electromagnetic wave in curved spacetime
I am trying to calculate the stress–energy tensor of an electromagnetic wave in curved spacetime, characterized by the diagonal metric
$$
g_{\mu\nu} =
\begin{pmatrix}
-g_{zz} & 0 & 0 & 0 \\...
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Degrees of freedom in stress-energy tensor
The stress-energy tensor has 16 components, but this question is only about the 9 components $T^{ij}$ with $i,j=1,2,3$. According to Wikipedia, these components are defined as follows:
The components ...
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Showing that Poynting’s theorem preserved with Proca Lagrangian
The Proca Lagrangian is
$$\mathcal{L}=-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\mu_0\Lambda^2}A_{\mu}A^{\mu}+A_{\mu}J^{\mu}$$
Where $\Lambda=\frac{\hbar}{m_{\gamma}c^2}$.
The symmetric energy-...
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Question on linear perturbations in cosmology
I've been studying clustering dark energy when I came across a paper named "A Short Review on clustering dark energy" by Ronaldo Batista. there are 2 equations in this paper (eq.8 and eq.9) ...
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Pressure as momentum flux versus momentum of fluid flow
I've read in multiple sources (for example, Thorne and Blandford, Modern Classical Physics, pg 83 and Schutz, A First Course on General Relativity pg 92) that the stress components of the stress-...
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Why symmetrization of energy-momentum tensor doesn't add additional term to Lagrangian density? [duplicate]
I am self-studying the book “James H. Luscombe, Core Principles of Special and General Relativity”. In “CHAPTER 9 : Energy-momentum of fields” of the book, it starts by introducing Noether’s theorem ...
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