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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Proving the tensor virial theorem

In Schutz's Relativity Chapter 4, problem 23b) states: Use the identity $T^{\mu\nu} _{~~~~~,\nu} = 0 $ to prove the following results for a bounded system (i.e. a system for which $T^{\mu\nu} = 0 $ ...
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Deriving an equation from the energy-momentum tensor in general relativity

I have a mathy question regarding the Friedmann-Lemaître equations (FLE) in standard cosmology. For reference the FLE are as follows $$\frac{3}{R^2}\Big(k + \frac{\dot{R}^2}{c^2}\Big) = \frac{8\pi G}{...
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Why is trace of energy-momentum tensor not a good candidate for the source of gravity?

The reason why I think trace of energy-momentum tensor can't be the source is that, in a static weak field $p$<<$\rho$ $$T^{\mu}_{\mu} \approx T^{0}_{0} = -\rho$$ and if in one system I have ...
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Does it make sense to discuss the “pressure” of the universe?

I've heard the galaxies of the universe analogized as particles in a gas. If we consider this analogy, and understand that the universe is expanding while the temperature at this point in time is ...
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Change in stress tensor under a Weyl transformation

Let $T_{ab}$ be the stress tensor of a 2D conformal field theory with metric, $dzd\bar{z}$. If the same conformal field theory is defined on a manifold with metric $\Omega(z,\bar{z})^{-2}dzd\bar{z}$, ...
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How to define kinetic energy and potential energy from EM tensor in newtonian physics?

The question arises from here. People wants to define kinetic energy and potential energy from EM tensor. My question: How to define kinetic energy and potential energy from EM tensor in newtonian ...
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How to find the Hamiltonian from the energy-momentum tensor for a free electromagnetic field?

This question is related to a previous question that I have asked before titled: Energy-Momentum Tensor for the Electromagnetic Feild asking why the energy-momentum tensor had the following form $$T^{...
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Are the concepts of kinetic energy, potential energy etc not valid in general relativity?

In Newtonian physics, we come across different forms of energy, such as kinetic energy, potential energy etc. But in general relativity, we find only the total energy that is obtained from the energy-...
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Energy-Momentum Tensor for the Electromagnetic Feild

Question When calculating the hamiltonian for the free Electromagnetic Field with Lagrangian density $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ Using Noether's theorem I found the answer to be $...
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Does a non-lagrangian field theory have a stress-energy tensor?

In classical field theory, the stress-energy tensor can be defined in terms of the variation of the action with respect to the metric field, or with respect to a frame field if spinors are involved. ...
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Is polarization of light preserved in gravitational lensing?

Space should not have torque since the stress energy tensor is symmetrical. That would imply that gravity can not turn a polarization plane of an electromagnetic wave. Have any changes of ...
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Energy-momentum tensor for a Lagrangian with no explicit spatial dependence

Suppose I have a Lagrangian $\mathcal{L} = \mathcal{L}(\phi, \partial_\mu \phi)$ for a (let's say) real scalar field with no explicit temporal or spatial dependence. Then I believe the usual Noether ...
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Trace of the energy-momentum tensor

I had to find the canonical energy-momentum tensor defined by this Lagrangian density $ \mathcal{L} = - {1 \over 4} F_{\mu \nu} F^{\mu \nu}$ and I got the result of $ T^{\mu \nu} = - F^{\mu \lambda} \...
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The variation of the Lagrangian density for the canonical energy-momentum tensor

I expanded the Lagrangian to this form $$ \mathcal{L} = -{1 \over 4} F^{\mu \nu} F_{\mu \nu} = ... = - {1 \over 2} (\partial^{\mu} A^{\nu} \partial_{\mu} A_{\nu} - \partial^{\mu} A^{\nu} \partial_{\nu}...
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Energy-momentum tensor and gravity [closed]

Calculating from a given action the energy-momentum tensor $ \tilde{T}_{\mu \nu} $ (differentiating respect to $ \delta g^{\mu \nu}) $ I can create gravity by a generalization of the Einstein field ...
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What is the unit of Klein Gordon field?

Normally I don't care about units in the derivations on relativity or QM. Just set $\hbar = c = 1$. But learning about the energy momentum tensor for the Klein Gordon equation, I couldn't make $T^{00}$...
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Relation between bulk Hamiltonian in AdS and stress energy of CFT

Consider the following two situations: One can define a stress energy for AdS which matches with the expectation value for the CFT stress tensor. Consider bulk metric perturbations of the form: $$g_{...
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1st order approximation to energy-momentum tensor of gravitational field

I was studying linearized gravity and this approximation was given without any derivation. It might be clear for others but I'm quite new on GR and I'm not sure how to get this first order ...
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The conditions under which a stress tensor $T^{\mu\nu}$ exists

I used to believe that the existence of the stress tensor in a QFT has to do with translation invariance: "If a theory is translation invariant, then one can construct a conserved $T^{\mu\nu}$ by ...
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Liouville CFT Poisson Brackets

I have been given an action of the form: $$S = \frac{1}{4\pi}\int d^2\sigma \ \sqrt{-g}\left(\frac{1}{2}\partial_\mu\phi \partial^\mu\phi + \frac{1}{\zeta}\phi R + \frac{\mu}{2\zeta^2}e^{\zeta\phi} \...
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What's pressure in the stress-energy tensor? [duplicate]

This is a question about Einstein's field equations. In Einstein's field equations, there's a term called the stress-energy tensor. The 3 elements of the diagonal of the tensor are pressure terms. ...
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Constant term in action term in general relativity?

Question So I recently pondered the following. Let's say I have an $2$ actions $S_1$ and $S_2$ which differ by a constant: $$ S_1(\dot x_i, x_i) = S_2(\dot x_i, x_i) + \tilde c$$ Now their equations ...
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Understanding where all the equations from a covariant derivative of a tensor come from

Suppose I have a situation where i know that $\nabla_i T^{ik}=0$ where $ T^{ik}$ is a tensor of rank 2, which is diagonal, such as the perfect fluid energy-momentum tensor. We are dealing in a ...
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Straight cosmic string energy-momentum tensor and the cosmic strings EoS

Consider a simple infinite straight "cosmic" string of negligible thickness, in flat spacetime. The string energy-momentum tensor has the following components (in the string proper frame, ...
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Check that $\eta^{\mu\nu}T_{\mu\nu}=0$ [closed]

I'm trying to solve a problem and I have some problems for that. The problem is to show that $\eta^{\mu\nu}T_{\mu\nu}=0$, where $\eta^{\mu\nu}$ is the Minkoswki metric: diag(1,-1,-1,-1), and $T_{\mu\...
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What is the velocity $u^\mu$ in the stress-energy-tensor of a perfect fluid?

I am currently learning about fluid dynamics in special relativity. We defined the stress-energy-tensor of a perfect fluid to be \begin{equation} T^{\mu \nu} = (\rho + P) u^\mu u^\nu + P g^{\mu \nu}. \...
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Scalar field energy density

Considering a classical scalar field theory, I can find the canonical energy momentum tensor and if I calculate the $00$ component I get: $$T^{00}= \frac{1}{2} \dot \phi^2 + \frac{1}{2} (\partial_i) \...
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Covariant Derivative and energy momentum tensor

In this reference https://arxiv.org/abs/hep-th/0307199 pag.60, it is said that it is possible to find an infinitesimal spacetime diffeomorphism (a vector field) $X_{\nu}$ independently to its ...
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Einstein field equation specific solution

Do Einstein's field equations admit a solution such that spacetime was empty in the past of a hypersurface of constant time say $t =0$, but in the future there exists a non-vanishing energy momentum-...
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Equation in Field Quantization Greiner

Hello, I dont understand the red part. Shouldn't it have minus instead? Sorry for lack of formatting, I only have a phone.
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Spin part of the angular momentum from the Lagrangian

For fermions of spin $1/2$ the angular momentum has following form: $$ \mathcal{J}_z = \int d^{3}x \ \psi^{\dagger} (x) \left[i(- x \partial_y + y \partial_x) + i\sigma^{xy} \right] \psi(x) $$ Here ...
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Continuity equation for the stress-energy tensor in the FLRW metric

I'm trying to compute the continuity equation for the stress-energy tensor $\nabla^\mu T_{\mu\nu}$ in the FLRW metric $$ds^2=-dt^2+a^2(t)ds^2_3$$ where $ds^2_3=g^3_{ij}dx^idx^j$ is the metric for the ...
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Why are shear-stress and momentum-flux the same in the GR?

I am investigeting the meaning of the components of the Stress-Energy tensor: My source also states, that this matrix is always symmetric in the General Relativity. That looks obvious on the image - ...
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Covariant conservation of the energy-momentum tensor

In order to derive the geodesic equation of motion from the covariant conservation of the energy-momentum tensor we have to do the following procedure: $$ T^{\mu\nu}_{\space\space\space\space;\mu}= \...
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Energy-momentum tensor of a string

I was asked to compute the energy momentum tensor of a string whose Lagrangian is: $$\mathcal{L} = \frac{1}{2}\rho \dot{y}^2-\frac{1}{2}\tau y'^2 \tag{1}$$ I've used: $$T_{\mu\nu} = \frac{\partial\...
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Are these conservation laws always true?

Imagine a system of particles with the internal force on $i^{th}$ particle due to $j^{th}$ particle being given as $f_{ij}$ From the derivation of law of conservation of momentum and law conservation ...
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How to get Levi-Cevita symbol in the derivation for angular momentum using Noether's theorem? (David Tong Ex Sheet 1 Q6)

Working through David Tong's sheet here https://www.damtp.cam.ac.uk/user/tong/qft/oh1.pdf and can't follow how to get the Levi-Cevita symbol out the front? Its equation 15. I was looking at trying to ...
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Euler's equation from energy-momentum tensor conservation

Given the energy-momentum tensor for a perfect fluid: $$T^{\mu\nu}=(\rho+p)u^{\mu}u^{\nu}+pg^{\mu\nu}\space\space\space\space\space(1)$$ I was trying to obtain Euler´s equation: $$ (\rho+p)u_{\lambda;\...
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Radially-dependent stress-energy-tensor(?)

When we solve the Einstein Field Equations $G_{\mu\nu} = 8\pi T_{\mu\nu}$ one way of doing it is by specifying a symmetry (and thus a general form of the metric) and then specifying the stress-energy-...
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Symmetric energy-momentum tensor

In field theory, there's no guarantee that the energy-momentum tensor resulting from Noether's theorem is symmetric. The usual trick to construct a symmetric tensor is to add to the original energy-...
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How does one take the Divergence of a Tensor?

I’m trying to understand the derivation of the Maxwell stress tensor in Heald and Marion. Im confused how they go from 4.101 to 4.102 in the image above. I can't seem to see how 4.101 is the ...
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How did Einstein know where to put individual elements of E-M Tensor $T$ w.r.t. the corresponding tensor $G$? where $G=\kappa T$

We know the well known relation in General Relativity. $G=\kappa T$ Where G is the Einstein Tensor and T is the Energy-Momentum Tensor and K is the constant. I wanted to ask how did Einstein got to ...
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Energy stored in the curvature of spacetime?

Lets assume there are particles with mass. If the mass is big enough, they will accelerate toward the center of mass. Thus the particles gain kinetic energy and hence the total mass gets bigger. Since ...
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Physical meaning for the determinant of the stress energy tensor?

I wonder if there is any physical meaning associated to the determinant of the stress energy tensor. Or do we know at least some context in which this quantity is meaningful?
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Does the scalar curvature vanish whenever the stress energy tensor is traceless?

Assuming the cosmological constant is zero, the Einstein equations are: $ R_{\mu \nu} -\frac{1}{2} R g_{\mu \nu} = 8 \pi T_{\mu \nu} $. Using the definition of the scalar curvature, we may obtain: $ R ...
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Consequence of stress-energy tensor vanishing on-shell?

Let's say we have a QFT, which off-shell has a non-zero stress-energy tensor, but vanishes when the equations of motion are applied. If the stress-energy tensor vanishes off-shell, then all field ...
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Stress-Energy tensor for dust

Summary:: I find two different expressions for the EM tensor for dust, and both derivations seem right to me. Given the action for a system of dust $$S =-\sum m_q \int \sqrt{g_{\mu\nu}[x_q(\lambda)]\...
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Stress tensor force equilibrium in Feynman lecture book

I was reading about tensors and especially stress tensors in Feynman lectures on physics Which is on the webvsite enter link description here While reading the “31-6. The tensor of stress”, I was a ...
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How does one add matter coupling terms to the linearized Lagrangian for General Relativity?

In Spacetime and Geometry, Dr. Carroll provides a Lagrangian for Einstein's equations in vacuum assuming that the metric can be written in the form $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. The ...
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Why is there an extra term in definition of Noether current for spacetime translations?

I am reading Schwartz's Quantum Field Theory textbook. In chapter 3, Schwartz first defines the conserved current for a symmetry $\phi \rightarrow \phi + \delta \phi$ that depends on a parameter $\...

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