Questions tagged [stress-energy-momentum-tensor]

A rank-2 tensor in relativity, which expresses the flux of energy-momentum along timelike and spacelike axes. Also known as the energy-momentum tensor. In the Einstein field equations, it is the source of gravitational fields.

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Derivation of transformation law for the Hilbert Stress energy tensor

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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Einstein equation for Einstein-Maxwell theory [duplicate]

In the Black Hole Lecture Notes by Reall, In Section 6.1 it is written that The action for Einstein-Maxwell theory is $$S = \dfrac{1}{16\pi} \int d^4x \sqrt{-g} (R - F_{ab}F^{ab})$$ where $F = dA$ ...
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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 ...
PhysicsIsHard's user avatar
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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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How we can deduce from symmetry of $ θ^{μν} $ that total energy-momentum due to field spin is zero?

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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How to mathematically describe the process of spacetime curvature?

I guess as a result of the energy-momentum tensor $T_{\mu\nu}$ coupling to a flat Minkowski metric, $\eta_{\mu\nu}$, the flat metric can become that of a curved spacetime, $g_{\mu\nu}$. How can one ...
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Inconsistency in Virasoro expansion of stress energy

As explained in Axiom 2.3 on page 7 of https://arxiv.org/abs/1609.09523, the independence of the stress tensor $$T(y)=\sum_{n}\frac{L_n}{(y-z)^{n+2}}$$ on the choice of expansion point $z$ leads to ...
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Is conservation of energy a local law in Quantum field theory? [closed]

From Wikipedia, "The local energy conservation in quantum field theory is ensured by the quantum Noether's theorem for the energy-momentum tensor operator. Thus energy is conserved by the normal ...
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Is there an intrinsic energy-momentum associated with constraining forces that don't do work?

As an example, consider a continuous charge distribution, within Maxwell's model of classical electrodynamics, that is brought from infinity onto a spherical surface at a radius $r$ from the origin. ...
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Showing that derivative of energy-momentum tensor is equal to 0

Given, \begin{equation} T^{\mu\nu} = F^{\mu\lambda} F^\nu{}_{\lambda} - \frac{1}{4} \eta^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}. \end{equation} Here $(T^{\mu\nu})$ is the energy-momentum ...
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Norm for the energy-momentum vector: what meaning/use does it have from the point of view of the energy-momentum tensor?

Many books on relativity define "mass" $M$ as the norm of the energy-momentum vector $\pmb{P} := (E, \pmb{p})$, that is, $M = \sqrt{\lvert \pmb{P}\cdot\pmb{g}\cdot\pmb{P}\rvert}$, where $\...
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Do negative pressures in Thermodynamics lead to a negative stress energy tensor?

If we have a gas or liquid described by the van der Waals gas law with negative pressure, does that lead to a negative stress energy tensor? Does a stretched liquid for example have a negative stress ...
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The meaning of the stress-energy-momentum tensor

I just learned some General Relativity and have a couple of questions about the stress-energy-momentum tensor $T$. In what follows, please let’s suppose that General Relativity and the Standard Model ...
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How does the amount of energy bound in the gravitational field of an object relate to the energy of the object?

If I understand correctly, there is energy bound in a gravitational field, although acceleration of the body that causes the field is required to release some of that energy (in the form of ...
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How should I calculate the commutator between the Belinfante stress tensor and the field operator?

As known, there is an ambiguity on the definition of the stress tensor (or energy-momentum tensor). The canonical stress tensor, defined as the Noether's current corresponding to space-time ...
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Thermal conductivity from the mass distribution of an object

Thermal conductivity is a property normally related to material. But it is also possible to relate it to objects. Consider two objects of different shapes made by the same homogeneous material. How ...
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Can you assume the energy-momentum tensor is symmetric if you only impose Lorentz symmetry?

The proof showing that the energy-momentum tensor is symmetric uses the fact that $\partial_\nu T^{\mu\nu}=0$ due to translation symmetry, the definition of the conserved current and that $\partial_\...
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Confusion about the derivation of stress tensor OPE from Ward Identity

I apologize for any difficulty in expressing my review. Allow me to briefly summarize the material and then pose my question. Review In David Tong's string lecture note, he derives the OPE between ...
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Canonical electromagnetic stress-energy-momentum tensor

I have canonical electromagnetic stress-energy-momentum tensor defined as: $T_{\mu\nu}=\frac{1}{4}\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}-F^{\mu\lambda}F^{\nu}_{\,\,\lambda}-F^{\mu\lambda}\...
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Q1.1(a) Sakurai Advanced Quantum Mechanics For energy-momentum tensor [closed]

I need to prove that the energy-momentum tensor density is defined as: \begin{equation} \mathcal{T}_{\mu\nu}=-\frac{\partial \phi}{\partial x_\nu}\frac{\partial\mathcal{L}}{\partial(\frac{\partial \...
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Energy-Momentum tensor in Classical Mechanics

I can compute Energy-momentum tensor in classical field theory using Noether's theorem and translation invariance of action, but I think I can't exactly calculate how to calculate same thing in ...
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Which version of the equivalence principle affects the coordinate dependency of the Landau–Lifshitz pseudotensor?

We know that the energy-momentum of gravity can be defined by a pseudotensor called the Landau-Lifshitz pseudotensor, which is coordinate dependent. In fact, the gravitational stress–energy will ...
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Cosmological perturbation theory and relationship to Taylor series?

In cosmological perturbation theory, it's hard to find papers that would expose the general principle to perturb physical quantities (metric, fluid pressure and density, speed...) up to the $n$th ...
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Electromagnetic field pressure

Wikipedia gives that maxwell tensor components have minus in the electromagnetic stress energy tensor https://en.wikipedia.org/wiki/Electromagnetic_stress%E2%80%93energy_tensor. That mean the ...
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What is the Noether stress-energy tensor?

The goal is to find the formula for the stress-energy tensor, of mass-energy field. The Poincare group's algebra generates it with 3+3+4 generators, the first 6 or the continuous Lorentz subgroup (Let'...
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Post-Newtonian Stress-Energy tensor

I am currently studying Michele Maggiore's book - 'Gravitational Waves: Volume 1: Theory and Experiments'. On pages 245 and 246, each order --- until the second order --- of the stress-energy tensor. ...
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Question about proof of Weinberg-Witten theorem

In proving the Weinberg-Witten theorem, there is a step where one needs to show \begin{align*} \lim_{k' \to k}\langle k, \sigma | J^{\mu} |k', \sigma \rangle &= \frac{q k^{\mu}}{k^0}\frac{1}{(...
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Derivation of the Conformal Ward Identity in Di Francesco et al

I am reading section 5.2.2. (titled The Conformal Ward Identity) from Conformal Field Theory by Di Francesco et al. The authors write \begin{align} \partial_\mu(\epsilon_\nu T^{\mu\nu}) &= \...
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Stress-energy-momentum tensor and potential energy

The stress-energy-momentum tensor in General Relativity includes a mass density terms, which is related to energy via $E=mc^2$. How does potential energy figure into this, since potential energy is ...
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How to derive WZW model’s energy-momentum tensor? The result is of course the Sugawara construction

I want to know how to derive WZW’s energy-momentum tensor. We know WZW action is $$ S_{WZW}[g]=\frac{k}{16\pi}\int d^2x Tr(\partial g^{-1}\partial g) - \frac{ik}{24\pi} \int_B d^3y \epsilon_{abc} Tr(h^...
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Detailed derivation of the energy-momentum tensor from the Maxwell Lagrangian [duplicate]

I have started studying QFT, and I am currently reviewing briefly on the classical field theory. I have come across the Maxwell Lagrangian given by $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}. $$ ...
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Local Supersymmetry and Space-time Metric

So following from Simple Supergravity (arxiv:2212.10044), on page 5, it's written that Only the spacetime metric can couple to the energy-momentum tensor... Can anyone explain why it must be the ...
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Transforming stress tensor from polar to Cartesian coordinates?

Trying to understand where my calculation goes wrong. I have the stress tensor with components $T_{\rho \rho}, T_{\rho \theta}$ and $T_{\theta \theta}$. I wish to express this in Cartesian coordinates....
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Lorentz Transformations and Angular Momentum unclear derivation

How do we get rid of $\omega^{\rho}_{\;\;\nu}$ and howt do we derive the last equation form the previous one. Here is URL for the source where I found this text page 17 (marked at bottom).
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Stress-Energy Tensor for a Two Mass System

I just don't understand this tensor and would like to go through an example with you to somehow make sense of it. I consider two spheres with masses $m_1$ and $m_2$, densities $\rho_1$ and $\rho_2$ ...
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First-order formalism General relativity

I am trying to find the stress energy tensor for an ideal fluid from a first-order formalism. My matter lagrangian is $\int d^4 \sqrt{-g} [g^{\mu\nu} \partial_{\mu} \theta \partial_{\nu} \theta - V(\...
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Stress-energy tensor spin-1 coupling

Would it be possible for the stress-energy tensor ($T_{\mu\nu}$) to couple to a spin-1 anti-symmetric rank-2 field (like the electromagnetic field strength tensor $F_{\mu\nu}$)? If so, what would be ...
physics_2015's user avatar
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Trace of stress-energy tensor for a scalar field

I'm trying to reproduce a calculation done in Birrell & Davies' book Quantum Fields in Curved Space (page 191). Given Klein-Gordon's equation $$(\Box + m^2 + R\xi)\phi = 0$$ and $$T_{\mu \nu} = (1 ...
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