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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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Do photons bend spacetime or not?

I have read this question: Electromagnetic gravity where Safesphere says in a comment: Actually, photons themselves don't bend spacetime. Intuitively, this is because photons can't emit gravitons,...
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In 2d CFT, why the $T_{zz}$ component of energy-momentum tensor is holomorphic even at quantum level?

In 2d Conformal Field Theory, the $T_{zz}$ component of energy-momentum tensor is treated as a holomorphic function $T(z)=T_{zz}$ at quantum level such as in OPE involved energy-momentum tensor. I ...
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Curvature created by an object near Earth via energy-stress tensor

From Misner... who uses the convention of (-, +, +, +) for the metric $g^{\mu\nu}$, with the electromagnetic stress energy tensor being(pg.141): $$T^{\mu \nu} = \frac{1}{\mu_0}(F^{\mu \alpha}F^{\nu}{...
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Imperfect or perfect fluid in Einstein Field Equation

I'm trying to solve the Einstein Field Equations in an unconventional way (at least not usual from what is done in most basic textbooks). So basically, I specified a metric tensor (specifying a ...
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Derivation of the Electromagnetic Stress-Energy Tensor in Flat Space-time

I am working on deriving the electromagnetic stress energy tensor using the electromagnetic tensor in the $(-, +, +, +)$ sign convention. However, I have hit a snag and cannot figure out where I have ...
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Energy-momentum tensor of the electromagnetic field

I have to derive the electromagnetic energy-momentum tensor from Noether's theorem and translation invariance. Due to translation invariance and gauge transformation: $$\delta A_\mu= a^\nu F_{\mu\nu}$$...
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Electromagnetic Stress-Energy Tensor in curved space-time

I found on Wikipedia that the electromagnetic stress energy tensor in curved space-time with sign convention $(-, +, +, +)$ is $$T_{\mu\nu} = -\frac{1}{\mu_0} \left ( F_{\mu \alpha} g^{\alpha \beta} ...
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General Relativity - (numerically) compute the metric from the stress-energy tensor?

I am new to GR and I am having trouble understanding how one goes back and forth between the metric $g_{\mu\nu}$ and the stress-energy tensor $T_{\mu\nu}$. First, have a look at the following post. ...
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Expectation value of descendant fields

I'm trying to calculate the following quantity: $ \left<(L_{-1}\phi)(w_1)(L_{-1}\phi)(w_2) \ldots (L_{-1}\phi)(w_N) \right>$ where $\phi(w_i)$ is a primary operator and $L_{-1}$ is the ...
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How to calculate the stress energy tensor of a particle of rest mass m?

I was trying to calculate the stress energy tensor of a point particle of rest mass m whose world line is given by $w^\mu(\tau)$ where $\tau$ is proper time. But I am not getting the correct answer. ...
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Electromagnetic stress-energy tensor to be used in Einstein's Field Equations

I am trying to put in the electromagnetic energy-stress tensor in for the energy-momentum tensor of Einstein's field equations. I am, however, unsure as to which tensor matrix to use. I found the ...
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$T \bar{T}$ OPE

In page 157 of Di Francesco (Conformal Field Theory) it is said that the holomorphic and antiholomorphic components of the energy-momentum tensor have the trivial OPE $T(z) \bar{T}(\bar{w}) \sim 0$....
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Is there a limitation on the values ​that Einstein tensor $G_{\mu\nu}$ can take?

Is there a limitation on the values ​​that Einstein tensor $G_{\mu\nu}$ can take? For example: Is it always bigger than zero? What is the highest amount that can be taken by it? What is the ...
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Diagrammatic expansion of an operator insertion in path integral for Trace Anomaly calculation

Starting with a scale invariant classical field theory, we can prove that the energy-momentum tensor will be traceless. \begin{equation} \Theta^\mu_{\ \mu }=0 \end{equation} In the context of the ...
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Relation between the trace anomaly and the energy-momentum tensor being off-shell

Let's say we have a massless QED theory with a Lagrangian \begin{equation} L=i\bar{\psi}\not{D}\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} \end{equation} The symmetric energy-momentum tensor is \begin{...
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Energy momentum tensor of EM field written in symmetric form

I'm reading A. Zee's book, Einstein Gravity in a Nutshell. In problem 7 of chapter IV.2, it is said that the energy momentum tensor of the electromagnetic field \begin{align} T^{\mu\nu}=\eta_{\lambda\...
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Integration by parts, Weinberg Cosmology p.526 [closed]

How do I perform this integration by parts done explicitly? $$0 = \delta I_m = \int d^4 \sqrt{-g} T^{\mu \nu} \left[- \frac{\partial \epsilon^\rho}{\partial x^\mu} g_{\nu \rho} - \frac{\partial \...
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Defining pressure from the stress-energy tensor components

Suppose I have a trial metric which when I plugged into the Einstein Field Equations produced a stress-energy-momentum tensor where the following components are non-zero: $T_{tt}$, $T_{tr}=T_{rt}$, $...
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Extending general relativity with torque based on quasimetrics

If torque is allowed to exist in the space part of the stress energy tensor $T_{\mu\nu}$ in the Einstein field equations $$ R_{\mu\nu}-\frac12 g_{\mu\nu} R = 8\pi T_{\mu\nu} $$ it would lead to $T_{\...
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Question on energy conservation from the stress tensor of a classical scalar field

I am struggling to answer an old general relativity exam question, which is as follows: "Consider a scalar field $\phi(t,x^i)$ with potential $V(\phi)$ on a general spacetime. Its stress tensor is ...
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Mass conservation in spherical coordinate

See four velocity $u^\alpha = \gamma(1,\beta,0,0)$ in a spherical coordinates $(ct,r,\theta,\phi)$, The mass conservation is \begin{equation} \nabla_\mu(\rho u^\mu) = 0 \end{equation} Then how it ...
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Stress-Energy Tensor and Conformal Invariance in String Theory

Since the Euler-Lagrange Equations corresponding to the Polyakov Action implies no dependance on the auxillary metric we arrive at the constraint $T_{ab}=0$. We then change to lightcone coordinates $++...
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Physically acceptable energy-momentum tensor

From a problem the line element is: $$ds^2 = -c^2e^{-2ax}dt^2 + dx^2+ dy^2+ dz^2$$ I found energy-momentum tensor ($T_{\mu\nu}$) from Einstein field equation by using the above line element. Only T$...
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Derivation of Holomorphic Ward Identities in Franceso's CFT

In equation 5.37 of francesco's CFT he writes the Ward Identities for traslation symmetry in the language of holomorphic functions. He goes from \begin{equation} \frac{\partial}{\partial x^\mu} \...
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Divergence energy tensor of Proca Lagrangian

Cosnider the Lagrangian $\mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu\nu}-\frac{m^2}{2}A_\mu A^\mu$. Then Euler Lagrange equations of motions are $\partial_\beta F^{\beta \alpha}-m^2 A^\alpha=0$. Then I ...
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How do you compute the stress-energy tensor for electromagnetism + gauge fixing term?

I want to compute the stress-energy tensor for the following Lagrangian: $$\mathcal{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} - \frac{1}{2\xi} (\nabla_\mu A^\mu)^2$$ but I'm struggling with the gauge-...
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How to Derive Energy-Momentum Tensors For Imperfect Fluid? [duplicate]

I was reading a book General Relativity: Introduction To Physicist and I found something Energy Momentum Tensor for imperfect Fluid but where can I find derivation of this?
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$\phi R$ term for scalar field in a curved background

Condider the following action for a free scalar field $\phi$ in a curved background $$S=\int dx\Big( \frac12g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi+\gamma \phi R\Big)$$ Here $g_{\mu\nu}$ is a ...
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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 ...