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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15 views

Deriving continuity equation from time component of $T_{\nu ; \mu}^{\mu}=0$

I know that $T_{\nu ; \mu}^{\mu}=0$. I want to derive the continuity equation from examining the time component in the Newtonian limit. I have $$T_{; \nu}^{t \nu}=\rho_{0} U^{\nu} U_{, \nu}^{t}+\...
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Motivation for tensor theory of gravity

In class we were shown that $$\rho = \frac{dm}{dV}$$ has the transformation properties of the 00 component of a rank 2 tensor. So we'd like to turn the classical Poisson equation for gravity into a ...
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If the lagrangian density changes by a total derivative of the lagrangian density

When we derive energy momentum tensor current by actively transforming field. We see that lagrangian ( density) changes by a total derivative of the lagrangian. If a total derivative of the function ...
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37 views

Shear viscosity in an acoustic plane wave

In the absence of bulk viscosity, $\eta_b = 0$, in Landau and Lifshitz book and many other places, the viscous stress tensor is defined as: \begin{equation} \sigma'_{ik} = \eta_s\left(\frac{\partial ...
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Why is the stress tensor simplifying like this

\begin{eqnarray} \nabla \cdot \boldsymbol \tau &=& 2 \mu \nabla \cdot \boldsymbol \varepsilon\\ &=& \mu \nabla \cdot \left( \nabla\mathbf{u} + (\nabla\mathbf{u}) ^\mathrm{T} \right)\\ &...
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Interpretation of equal absolute values of pressure and tension in the electric or magnetic field along Cartesian axes aligned with the field

For an electric or magnetic field along the $x$ axis, the stress-energy tensor in mixed covariant-contravariant form, in $(t,x,y,z)$ coordinates, is of the form $\operatorname{diag}(1,1,-1,-1)$ (...
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Sign of stress tensor in momentum conservation equation

I have a confusion and would be grateful if anybody could help clear it up. So I was reading Kip Thorne's modern classical physics and came up on the following definition of stress tensor: $T_{ij} $ ...
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71 views

Navier-Stokes Derivation

Someone knows where can I found a physical derivation of the Navier-Stokes equation? Mainly the stress tensor. A lot of authors simply "jumps" the stress tensor and it's the more important of physical ...
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Relativistic energy density (dust, presureless gas or number of particles)

The basic question on the relativistic energy density of the dust (pressureless gas or some number of moving particles). I assume that for some group of particles it is intuitively obvious to write ...
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662 views

A traceless stress energy tensor?

I'm trying to solve this exercise: Suppose an arbitrary theory (Flat space-time?) with a single field (Is a scalar field?) invariant under dilations, i.e. $x\mapsto b x$ and $\phi \mapsto \phi$. ...
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Generator of spatial translation in field theory

In classical mechanics, we know that the momentum operator is the generator of spatial translation. But it seems to me that this is no longer the case in the classical field theory. Lets first ...
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Confusion about stress-energy tensor in 2D gravity when reformulated as a CFT

I'm trying to verify that I obtain the same stress-energy tensor for a simple 2D gravity theory and the associated CFT reformulation. However, the results are not agreeing which leads me to believe I'...
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Stress-energy tensor of perfect fluid: changing reference system

The stress-energy tensor for a perfect fluid is given by, $$T^{\mu\nu}=\rho_0c^2h\frac{u^\mu u^\nu}{c^2}+pg^{\mu\nu}$$ where $h=1+\epsilon+\frac{p}{c^2\rho_0}$, $\rho_0$ the rest energy density, $p$ ...
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Steady flow of the relativistic dust - stress-energy tensor

Given relativistic dust (p=0) in flat spacetime. The dust moves in one direction along coordinate $x$ only. Consider this flow is steady. So for an observer at rest at fixed point of $x_0$ $v(x_0)$ ...
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Pressure in General Relativity

The standard definition of pressure (e.g. in wikipedia) is $p = \frac{F}{A}$, where $F$ is the magnitude of the normal force and $A$ is the area of the surface contact. In GR one sometimes also talks ...
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Energy-momentum Tensor for a Real Scalar Field Lagrangian

I'm currently working through Schwartz's QFT book, and I'm trying to find the energy-momentum tensor for the following Lagrangian: $$ L = -\frac{1}2\phi(\Box+m^2)\phi. $$ Am I correct in thinking ...
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Stuck Solving MTW Gravitation Problem 20.5

I am stuck on exercise 20.5 part a) from Misner, Thorne, and Wheeler's Gravitation chapter 20. The Einstein summation convention is used throughout this post. Problem Statement Calculate $t^{\...
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Should the energy-momentum tensor be invariant under gauge transformations?

For example, consider the electromagnetic theory given by \begin{align} I=-\frac{1}{4}\int d^4x\, F_{\mu\nu}F^{\mu\nu}, \end{align} where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. The action ...
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Can we use energy–momentum relation with scalars to get meaningful result?

Does energy–momentum relation (as defined in the beginning of wiki/Energy–momentum_relation) meant to be used only in terms of stress-energy-momentum tensor but not with simple scalars?
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Neumann Boundary Conditions for the Open string and the energy momentum tensor

I read in Polchinski's book, "String Theory", page 56, that for the open string the energy momentum tensor satisfies equation (2.6.26) at a boundary $$ T_{ab}n^a t^b=0 \,, $$ where $n^a$ and $t^a$ are ...
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Variation of the metric w.r.t. the metric in derivation of stress tensor

Consider massless free scalar theory $$S = \int d^4x \sqrt{-g}L = \int d^4x \sqrt{-g} \;g^{cd}\nabla_c\phi \nabla_d \phi.\tag{1}$$ To compute the Hilbert stress-energy tensor we require $\sim\delta S/\...
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$bc$-system energy momentum tensor

I have a (maybe silly) question regarding the expression of the energy momentum tensor of the $bc$-system in equations $(2.5.11a)$ and $(2.5.11b)$ in Polchinski's String Theory, page 50. I know that ...
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Physical significance of the canonical energy-momentum tensor

I have a question regarding the physical significance of the canonical energy momentum tensor $T_\nu ^\mu$ in the context of classical field theory. It is defined as $T_\nu ^\mu = \frac{\partial \...
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Riemann Dual Tensor and Scalar Field Theory

I'm trying to find the component equation of motion for the action in a paper. The action for the system is, $$S=\frac{m_P^2}{8\pi}\int d^4x\sqrt{-g}\bigg(\frac{R}{2}-\frac{1}{2}\partial_\mu\phi\...
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Stress energy pseudotensor of the repulsive gravity of an exotic matter

Postulating the existence of a hypothetical negative mass, would the pseudotensor of its repulsive gravitational field have a negative stress energy? If so, why?
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Frame-dependence of the gravitational field pseudotensor

What seems to be the common consensus in physics is that a gravitational field does not have a stress energy tensor due to the equivalence principle, but rather a pseudotensor. Is this pseudotensor ...
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How does the stress energy tensor change in different reference frames?

Is the Stress-Energy tensor invariant in all RFs? If not (which is highly probable) how does it change? EDIT: does the Einstein equation help? Since (without $\Lambda$) $$ R_{\alpha\beta} -\frac{1}{2}...
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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 ...