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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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Effect of Compression/Tension on the Rest Mass of an Elastic Solid

If I compress an elastic solid I will strain the material in the direction of the force and therefore do work that is stored as potential energy in the material. The resulting material will have no ...
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Energy momentum tensor in polarizable media

I am trying to follow Soper's derivation of the energy-momentum tensor for polarizable media in his book 'Classical field theory'. Given a Lagrangian density $\mathcal{L} = \mathcal{L}(\phi_A, \...
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Electromagnetic energy stress tensor with non zero current

I have read through Jackson and also Classical field theory (Florian Scheck) on this topic and neither of them address my question. In both of them we are told that the canonical stress energy tensor ...
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Is there a scalar field that gives rise to the energy-momentum tensor of a perfect fluid?

If I understand correctly, Sean Carroll's Spacetime and Gravity says that the energy-momentum tensor for a perfect fluid $$T^{\mu\nu} = (\rho + p)U^\mu U^\nu + pg^{\mu\nu}$$ can be obtained as the ...
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Why are the Klein-Gordon equations warranted from the conservation of the energy-momentum tensor?

If we have an action with a scalar field non-minimally coupled to the gravity: $$\int dx^4 \sqrt{-g}(-\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}\zeta R\phi^2-V(\phi)).....(1)$$ varying ...
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Maxwell stress tensor for electromagnetic wave

Sorry if this is a naive question but I've been struggling in trying to proof this for a week. Consider an electromagnetic wave with wave vector $\vec{k}=k\hat{n}$, the Maxwell stress tensor can be ...
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Question about explicit notation of averaged energy conditions integrals

Beyond the basics of general relativity, we rapid encounter the so called Averaged energy conditions. The mathematics of these quantities are related to line and volume integrals. As given by [1], ...
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Covariance of applying a four-force to a stress-energy tensor with forward Euler

I've run into a great deal of confusion on what I expected to be a very simple issue of covariance. I have an equation $$T^{\mu\nu}_{~~~~;\mu} = -G^{\nu}.$$ This is manifestly covariant; so far so ...
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Density $\rho$ in the Friedmann equations

In the Friedmann equations: $$\ddot{a}=-\frac{4}{3}\pi G(\rho+\frac{3p}{c^2})$$ $$\dot a^2+Kc^2=\frac{8}{3}\pi G\rho a^2$$ I didn't understand if $\rho$ is the mass density deriving from $m_0$ (the ...
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Fluid in spatially enclosed universe and some additional assumptions?

Background In general relativity I was wondering about the following Universe: Imagine a spatially closed universe containing only a perfect fluid: $$x'_i =x_i + L_i $$ where $i =1,2,3$ one of the ...
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What's the most general form of Maxwell's stress tensor for EM fields in matter?

In Griffith's Introduction to Electrodynamics we find a derivation of the relation $$\mathbf f + \frac{\partial \mathbf g}{\partial t} = \nabla \cdot \overleftrightarrow{\mathbf T} ,$$ where $\mathbf ...
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System's mass and holographic boundary

Can mass map onto a holographic boundary in AdS(or dS)/CFTs? In particular, might the mass of a system vary directly with the surface area of a characteristic holographic boundary? I'm guessing maybe ...
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Energy-momentum tensor of the Dirac field

I'm trying to compute the energy momentum tensor for the dirac field $$\mathcal{L}=\bar\psi(i\gamma_\mu\partial^\mu-m)\psi $$$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}\partial^...
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Energy of string in bosonic string theory

When we start from the Polyakov action, we can choose to work in the conformal gauge $h^{\alpha\beta}=\eta^{\alpha\beta}$ where $h^{\alpha\beta}$ is the metric on the world-sheet and $\eta^{\alpha\...
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Stress tensor of an elastic medium

I don't understand a passage from the book I'm reading about tensor analysis. The state of stress of an elastic medium can be expressed by the stress function $\mathbf{p}(r,n)$ so that the force ...
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Are particles in a perfect fluid in random motion?

A perfect fluid has no heat conduction, but it exerts pressure in all directions (according to stress-energy-momentum tensor). If it does not conduct heat, then it means it does not have random ...
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Should the parallel propagator appear in the point-split stress-energy tensor?

The first step in Hadamard regularization of the stress-energy tensor of a free Dirac field is to write out the point-split expression $$\langle T_{\mu \nu} \rangle \equiv \frac{1}{4} \lim_{x'\to x} \...
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Einstein and Riemann curvature tensor

Riemann curvature tensor is dirrectly related to a path dependence of parallel transport. I read that Einstein first thought of this tensor to be the one that goes into his field equation but it didn'...
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Matrix elements of stress-energy tensor $\langle q | T^{\mu\nu} |q\rangle$ in QFT?

In many QFTs we can define a stress tensor $T^{\mu\nu}$. What is the matrix element of $T^{\mu\nu}$ in momentum eigenstates? For instance, consider $$\langle q | T^{\mu\nu} |q\rangle$$ in QCD, ...
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Using Maxwell Stress Tensor to get force between two current-carrying wires

I’m trying to find the force per unit length between two parallel wires carrying the same current in the same direction and a distance of 2a apart. I need to use the Maxwell stress tensor and am ...
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When and how was the need for symmetry in the stress-energy tensor first realized

This question is somewhat historic. Let $\Theta_{\mu\nu}$ denote the canonical stress-energy tensor of some matter field $\psi$ in special relativity. It is often stated that the reason why we want ...
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Momentum in capacitor field; How can an EM field have zero momentum density but non-zero momentum flux?

Consider the case of a simple, stationary parallel plate capacitor oriented with its plates lying in the x-y plane. The E-field is simply given by: $$\vec{E} = \frac{Q}{\epsilon_0A}\hat{z} $$ with ...
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Sign mistake in the energy momentum tensor of the Klein-Gordon Equation

Recently I understood that the energy momentum tensor can be calculated by: \begin{equation} T_{\mu \nu}=\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{\mu \nu}}.\tag{1} \end{equation} So consider ...
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What kind of average could give a Lorentz invariant energy-momentum tensor?

The electromagnetic (EM) radiation energy-momentum tensor is of the following shape, in the case of incoherent superposition of EM plane waves (I'm using $c = 1$ to simplify things, and metric ...
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Confusion about conservation of angular momentum tensor in classical field theory?

In my lectures, we considered the conserved stress energy tensor $T^{\mu \nu}$ and noted that we could always add a conserved tensor to it such that $T^{\mu \nu}$ is symmetric. As a consequence, a ...
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Demonstration of magnetic field lines “tension” and “pressure” using the Maxwell stress tensor

I would like to show that the magnetic fields lines are under "tension" along the lines and exert "pressure" perpendicularly to the lines, using the Maxwell tensor only. I have a sign ambiguity. In ...
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How do compute an energy momentum tensor, given some equations of motion

This problem can be found in a paper called "Gravitational Radiation From Point Masses In A Keplerian Orbit", but I do not have access to this, so cannot see how to do it. I have been given a ...
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Angular momentum of a circualr polarized EM wave

In an exercise, I am being asked to compute the angular momentum of a circularly polarized wave. The wave is defined by the four potential: $$\Phi^\mu(x^\nu) = \text{Re} \left\{ \varepsilon^\mu e^{...
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Hanging two charged sphere by a light string

NB: This is not a homework question. I am not searching for any solution of a math problem. I found something incorrect to do always in the nature of two charged pith balls hanging from a light ...
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Constraints in general relativity

In this review on inflation, on Pg. 135, Baumann talks about the energy and the momentum constraints for gravity. Are these equations the $G_{00} = T_{00}$ and $G_{0i} = T_{0i}$ components of the ...
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Proof of the existence of the energy-momentum tensor [duplicate]

I have a problem providing or finding a general proof for this statement i found in Mussardo's statistical field theory book, section $10.3.2$: Due to the locality of the theory there exists a local ...
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Is it possible to define an energy momentum tensor for classical point particles from a QFT?

I have a question about the semi-classical limit of a QFT that so far I have never been able to solve. Let's start with a second quantized Klein-Gordon field with Lagrangian $$\mathcal{L}(\phi)=\frac{...
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Why are the diagonals of the pressure tensor non-negative?

I understand that the pressure tensor is simply the momentum flux which makes sense to me (pressure is force per unit area which is momentum change per unit time per unit area). From this, a simple ...
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Stress Tensor decomposition into Compression & Tensile forces

this might be a little more of an engineering question that physics but physics is my background so asking here... I'm working on a project that does a finite element simulation of a mesh. I can ...
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Absorbing the Cosmological “Constant” in the standard Energy-Stress Tensor

Recently I found some publications on Cosmologies with variable cosmological constant. The Bianchi Identity then implies that the divergence of the modified Energy Stress: Tensor $$\hat{T}_{ab}=T_{ab}...
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Double divergence of stress tensor for migration flux

I am looking to calculate migration as a function of time using equation in Image 1. SigmaP is the total particle stress tensor in the cylindrical coordinates (r, theta, z). I am only interested in ...
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Einstein GR and metric signature

Let us take the einstein Equation $R_{\mu\nu} -\frac{1}{2}g_{\mu\nu}R = T_{\mu\nu}$. I'm just ignoring all the constants. For a perfect fluid, $$T_{\mu\nu} = (\rho + P)u_{\mu}u_{\nu} - Pg_{\mu\nu}.$$ ...
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What does it mean if the Lagrangian density has explicit spatial dependence?

First off, I have seen this post here which asks seems to ask my question, but it is not properly answered. If the Lagrangian has explicit time dependence, then the total energy, and Hamiltonian, is ...
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How much of Maxwell's equations is recoverable from the zero divergence of the stress-energy tensor?

As a motivating example, consider the static electromagnetic field defined by $\textbf{E}=(\text{const})x\hat{\textbf{y}}$, $\textbf{B}=0$. The stress-energy tensor for this field is $T=\operatorname{...
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Conservation of improved energy momentum tensor of a real massless scalar field

So I'm supposed to find that the improved energy momentum tensor of the scalar field $\phi$ satisfying the evolution equation $\Box \phi = 0$ is conserved. The improved energy momentum tensor is: $T^{...
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Noether charges of spacetime translation in KG field

When applying a spacetime translation $x^\mu\rightarrow x^\mu+a^\mu$ the KG lagrangian density changes by - $$\mathcal{L} \rightarrow \mathcal{L} + a^\nu \partial_\mu \delta^\mu_{\;\nu} \mathcal{L}$$...
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What is the geometric interpretation of the Einstein tensor $R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R$

The Riemann curvature tensor $R_{\mu \nu \rho \sigma}$ has the geometric interpretation of giving how much parallel transport fails to close around tiny loops. The Ricci tensor $R_{\mu \nu}$ the ...
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Stress-Energy Tensor of Electromagnetic Field with sources

I can find a lot of references which treat the derivation of Maxwell equations and the associated Energy-Stress Tensor from the action principle. But I cannot find any information on the Energy-Stress ...
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Divergenceless of energy momentum tensor for any metric $g_{\mu\nu}$

As suggested by @my2cts, from this post, I want to know if the divergenceless of energy-momentum energy tensor is valid for any metric $\eta_{\mu\nu}$ (i.e for example with $\eta_{\mu\nu}=g_{\mu\nu}$)?...
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Sign problem in electromagnetic stress energy tensor

I'm having a silly problem in calculating the electromagnetic stress energy tensor: the Lagrangian is $$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} $$ and the stress energy tensor reads $$ T^{\mu\...
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Get relation from definition of stress-energy tensor and the conservation of energy

Starting from the following definition of stress-energy tensor for a perfect fluid in special relativity : $${\displaystyle T^{\mu \nu }=\left(\rho+{\frac {p}{c^{2}}}\right)\,v^{\mu }v^{\nu }-p\,\eta ...
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building expression for momentum from stress energy tensor

Let's say I have a stress-energy tensor with the the following non-zero components: the diagonal components $T^{00}, T^{11}, T^{22}, T^{33} $ and $T^{10}=T{01}$. I know that the energy density is just ...
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Schwarzian derivative from conformal factor

Suppose I have a 2D Lorentzian conformally flat metric $$ ds^2 = -\Omega(u, v) du dv.$$ I consider a conformal field theory whose stress-energy tensor $T_{ab}$ is known on the flat metric $$ds^2 = -...
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non-zero divergence of stress energy tensor

are there spacetime metrics where $\nabla_{\nu}T^{\mu\nu} \ne 0$ ? if the divergence of the energy momentum tensor is non-zero, what does that tell us about the spacetime aside from the fact that ...
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Transformation of the Stress-Energy tensor [closed]

My question is related to this one. However in my case, the Lagrangian can depend on higher order derivatives (so the second point made doesn't hold). Can someone help me with it?