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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Translational Ward Identity

The Ward identity corresponding to energy-momentum conservation (translational invariance) is (see for instance Di Francesco Eq.(4.63) ) $$\partial_\mu \langle T^\mu_\nu X \rangle = - \sum_i \delta(x-...
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Large $c$ limit and connected correlation functions in $2d$ QFT

EDIT: This question has been edited thanks to a comment. One of my definitions was wrong, so I have rewritten the whole question. I was reading this paper about $T \bar{T}$ deformations of $2d$-QFTs ...
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Timelike and spacelike projections in General Relativity and associated conservation laws

For any timelike curve $p_\mu$ in General Relativity (section 3 of this review), we can project this into its timelike and spacelike components. Further, these projections are associated with ...
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Why is the stress-energy tensor for electromagnetic radiation traceless?

A photon gas obeys the equation of state $\rho=P/3$ and hence $T^{\mu}_{\quad\mu}=3P-\rho=0$. (Can also be seen by expressing the stress energy tensor in terms of of the electromagnetic tensor as ...
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Equation of Motion for a Test Particle

I have a question regarding Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973), Gravitation ISBN 978-0-7167-0344-0. It is a book about Einstein's theory of gravitation. At the page ...
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Deriving the stress-energy tensor from the Einstein-Hilbert action

I'm a mathematician who knows very little physics and is trying to learn relativity theory from a mathematical perspective. Let $M$ be a compact, orientable manifold. In the vacuum, the Einstein-...
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How electromagnetic energy-momentum looks like for arbitrary 4-velocity vector?

I need to expresse the electromagnetic energy-momentum tensor in a vacuum $$T^\nu_{\ \ \ \mu} = F_{\mu\alpha}F^{\nu\alpha} - \frac14 F_{\alpha\beta}F^{\alpha\beta}\delta^\nu_{\ \ \mu}$$ in terms of ...
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How do I measure pressure inside a water balloon?

I have fill a ballon with water. The balloon becomes big for its elasticity. So a potential energy is been stored on the surface of the baloon that is giving pressure on the water inside. Now I need ...
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Action & Energy-Momentum Tensor for Matter Fields

Pg 163 of "Tensors, Relativity and Cosmology" The action integral of a given matter distribution can be written in the form $$I_K=-c\int_\Omega\frac{\rho}{\sqrt{-g}}\frac{ds}{dt}\sqrt{-g} d\Omega ...
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On what is the pressure in Relativity exerted?

In Relativity we have the stress-energy-momentum(-pressure) tensor: The three green entries represent three pressures (in this extensive article much is said about the pressure term, but I still don'...
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What is the CFT dual of the stress tensor in the bulk?

I am new to AdS/CFT. I know that the dual of the bulk metric is the CFT stress tensor but what about the dual of the bulk stress tensor? I mean in principle one can extrapolate whatever bulk fields to ...
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Some aspect of covariant derivative of point particle energy-momentum tensor

My question is related to Derivation of the geodesic equation from the continuity equation for the energy momentum tensor I need to understand one step in derivation. Let's consider the Energy-...
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Covariant derivative in a basis

Reading through this paper, I saw that the energy momentum conservation: $$\nabla_\mu T^{\mu\nu}=0$$ can be evaluated as: $$\partial_t(\sqrt{-g}T^{t}_\nu)=-\partial_i(\sqrt{-g}T^{i}_\nu)+\sqrt{-g}T^...
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Noether current Lorentz rotation massive vector field

I'm considering a massive vector field in classical field theory. With the Lagrangian density $$\mathscr{L}=-\frac{1}{4}V^{\mu\nu}V_{\mu\nu}+\frac{1}{2}m^2V^{\mu}V_{\mu}.$$ I want to prove from the ...
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Transformation of the energy-momentum tensor under conformal transformations

I am reading the yellow book of Di Francesco about conformal field theory, and there is a step that he takes that I cannot follow while deriving the transformation law of the energy-momentum tensor ...
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Unorthodox way of solving Einstein field equations

Usually when we solve field equations, we start with a stress energy tensor and then solve for the Einstein tensor and then eventually the metric. What if we specify a desired geometry first? That is, ...
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Under what conditions is field momentum an eigenvector of the stress-energy tensor?

This question is vaguely related to an earlier question, but is more focused. In a paper, Electromagnetic Mass, Charge, and Spin, which apparently is not peer-reviewed, the author relates a "matter ...
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Contravariant Stress Tensor: expression derivation

I was trying to go through a paper on relativistic, viscous radiation hydrodynamics, when I came upon this expression, for the contravariant stress tensor: $T^{\alpha\beta}=\rho h u^\alpha u^\beta +P ...
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Derivation of Maxwell's Equations using the Energy-Momentum tensor [duplicate]

If the energy momentum tensor is related to the EM field tensor by $$ T^{\mu v}=F^{\mu \sigma}F^v_\sigma-\frac{1}{4}\eta^{\mu v}F^{\sigma \tau}F_{\sigma \tau} $$ Is it possible to derive Maxwell's ...
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Adding a total derivative to the Lagrangian does not preserve $\int\mathrm{d}^3\mathbf{x}~ T^{00}$

In problem 3.3 of Schwartz's QFT, the first two questions ask us to prove that if we add a total derivative to the Lagrangian: $$ \mathcal{L}\mapsto\mathcal{L}+\partial_\mu X^\mu\tag{1} $$ then $$ \...
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Traceless stress tensor

What does it mean, when the viscous (or viscoelastic) stress tensor is traceless $\tau_{rr}+\tau_{\theta \theta}+\tau_{\phi \phi}=0$? Why if the viscoelastic model is linear it is traceless and if ...
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Energy-momentum for gravity [duplicate]

My question is as follows: Why is it problematic to define energy-momentum tensor for the gravitational field? P.S. It is well-known that in GR we get the energy-momentum tensor of "matter" by ...
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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 ...
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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 ...