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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Dimensional inconsistency in first law of black hole thermodynamics

The first law of black hole mechanics (let's simplify by considering a uncharged and non-rotating black hole) can be written as $$\delta M = T \delta S$$ If I use the definition of Hawking ...
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41 views

“Simple” Variation of the gravity action with boundary

I'm concerned with the derivation of the quasi-local stress tensor (getting from eqn 2.4 to eqn 2.6 in this paper: http://arxiv.org/abs/hep-th/0508218). As is the case with all the references I have ...
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78 views

Boundary stress-energy tensor form ADS/CFT

In "Gravitational Dynamics From Entanglement "Thermodynamics"" by Lashkari/McDermott/Van Raamsdonk, the authors derive the linearised Einstein equations from ADS/CFT. At page 6 they use ...
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Avoiding Pseudo-tensors when addressing global conservation of energy in GR

Discussions about global conservation of energy in GR often invoke the use of the stress-energy-momentum pseudo-tensor to offer up a sort of generalization of the concept of energy defined in a way ...
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39 views

Existence and uniqueness of solutions to $\nabla^a T_{ab}$ in general (or special) relativity

The equation in the title of this question can be a relativistic analogue of the Navier-Stokes equation (in the sense that, in the low-velocity limit, it reduces to Euler's equation when $T_{ab}$ is ...
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The question about MTW 4-momentum integral expression and lorentz nature

In section 5.8 of Misner, Thorne, and Wheeler's "Gravitation" there is a proof that 4-momentum determined as $$ \tag 1 p^{\mu} = \int T^{\mu 0}\,\mathrm{d}^{3}\mathbf r , \quad \partial^{\mu}T_{\mu ...
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70 views

Property of stress-tensor in flat spaces

Let $T_{ab}$ be a stress-tensor in a flat space satisfying conservation equations. Define $$ P^i=\int T^{oi}d^3x, \;\; D^i=\int T^{00}x^id^3x $$ Can anyone show me how to prove $$ \frac{dD^i}{dt}=P^i ...
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74 views

Non-locality of gravitational energy

Gravitational energy is non-local which is essentially because of the equivalence principle. The equivalence principle says that you can always transform your frame so that you feel like in a ...
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44 views

Quasilocal stress tensor

I have been reading through the paper hep-th/9902121 and have a few questions about the first five lines of the introduction: 1) "In a generally covariant theory, it is unnatural to assign a local ...
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82 views

Non-local gravitational energy tensor

The well-known derivation of the Landau-Lifshitz gravitational energy pseudotensor, relies on several requirements: 1) that it be constructed entirely from the metric tensor 2) that it be index ...
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146 views

Energy-momentum tensor

I need to show that: \begin{align} \mathcal h_i^a \, T_{ab} \, h_i^b=(\nabla_i \phi)^2-\frac{h_{ii}}{2}[\dot{\phi}^2-(\nabla \phi)^2-m^2 \phi^2] \end{align} where i) $T_{ab}=\nabla_a \phi ...
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83 views

Is there a general formula to translate from *canonical* to *physical* momentum?

In Peskin and Schroeder, after having derived a conserved tensor $T^{\mu \nu}$ associated with translations in space-time (the stress-energy tensor), it is said that the charges $\int d^3 x T^{0i}$: ...
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121 views

Applying Weak Energy Condition for a specific energy-momentum tensor

So, I have a particular energy-momentum tensor, for a specific line element, and I want to check if this obeys the weak energy condition ($T_{ \mu \nu} U^\mu U^\nu \geq 0$ where $U^\mu$ and $U^\nu$ ...
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161 views

Stress Energy Tensor of EM Field

Stress energy tensor for electromagnetic field is given by $$T^{\mu\nu}=\frac1{4\pi}(F^{\mu\alpha}F^{\nu}{}_\alpha-\frac14 g^{\mu\nu} F_{\alpha\beta}F^{\alpha\beta}).$$ My textbook (unpublished ...
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101 views

How to prove the energy of gravity in general relativity is non-local?

Every textbook in general relativity containing the energy of gravity all says that the energy of gravity is non-local and every energy-momemtum density received is pseudo-tensor, but "having not ...
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54 views

How to show isotropy of $SU(2)$ Yang Mills stress energy tensor?

When I vary the action of the YM Lagrangian density $$L = -\frac{1}{4} F^a_{\mu \nu}F^{\mu \nu}_a + J_a^\mu A^a_\mu$$ with respect to the metric, I obtain: $$T_{\mu \nu} = \frac{-2}{\sqrt{|g|}} ...
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234 views

Constructing Ward identity associated with conserved currents

Consider constructing the Ward identity associated with Lorentz invariance. It is possible to find a 3rd rank tensor $B^{\rho \mu \nu}$ antisymmetric in the first two indices, then the stress-energy ...
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56 views

stress tensor in Kazama-Suzuki construction

This is a technical question about equation (2.42) of the original paper [KS] of the Kazama-Suzuki construction. I think the authors did a simple substitution ...
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220 views

Solving the equation of relativistic motion

How does one solve the tensor differential equation for the relativistic motion of a partilcle of charge $e$ and mass $m$, with 4-momentum $p^a$ and electromagnetic field tensor $F_{ab}$ of a constant ...
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25 views

Inequivalent matter actions with the same stress-energy tensor in general relativity

In general relativity, suppose as usual that we have the following action for the matter fields \begin{equation} S_{\mathrm{matter}} = \int_M d^4 x \sqrt{-g} L_{\mathrm{matter}} , \end{equation} ...
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39 views

Imaginary part of the stress-energy tensor

I just encountered in an article of D. G. Boulware ("Quantum Field Theory in Schwarzschild and Rindler Spaces", Phys. Rev. D 11, 1404, 1975, in the last paragraph of the Introduction) the statement ...
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42 views

Can nonconserved energy in GR be thought of as going into gravitational field energy?

One of the most striking features of GR is that energy is not conserved. Carroll's GR text has an interesting statement about this: Clearly, in an expanding universe... the background is changing ...
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28 views

Stuck with physics tensile strength problem

So this was a question in my exam. I am not asking for the solution rather some help at a step. A glass spherical shell of radius $R$ has a point source of monochromatic light of wavelength ...
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79 views

Normal of a null surface and null junction conditions in general relativity

I am trying to use the null junction formalism in general relativity (as explained in eg http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.3763&rep=rep1&type=pdf, "Junctions and thin ...
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49 views

Calculate energy-density of known plasma in microwave?

Let's assume I want to create a plasma in a regular household microwave similar to this home-made experiment. Although I am dealing with a small amount of mass, I assume that the addition of microwave ...
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37 views

Is energy-momentum of curvature a boundary/holographic density?

Since the beginnings of General Relativity, we have had this awkward, unholy separation of the universe in marble versus wood. divergence of the stress-energy momentum holds at all points of ...
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92 views

unknown stresses in double-layer glass window

I live in cold place where outside temperature drops to -20. Currently, we have -20 and on my window, which is doubled layer glass with trapped air in between, I found a "polarized stress spectrum" ...
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49 views

Isolating the divergences in the stress energy tensor

In DeWitt's report "Quantum Field Theory in Curved Spacetime" (B. S. DeWitt, Phys. Rep. 19C, 292 (1975)), he states that in Eq.(175) $$\langle in, vac| T^{\mu\nu}|in,vac\rangle = 2 \frac{\delta ...
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228 views

Derivation of correction to canonical stress energy tensor due to addition of total divergence to Lagrangian

It is mentioned in almost every text book that equations of motions are not modified if we add a total divergence of some vector $$\partial_\mu \ X^{\mu}$$ to Lagrangian but canonical stress energy ...
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78 views

Constructing conserved current given the lagrangian

Consider the following Lagrangian for a massive vector field $A_{\mu}$ in Euclidean space time: $$\mathcal L = \frac{1}{4} F^{\alpha \beta}F_{\alpha \beta} + \frac{1}{2}m^2 A^{\alpha}A_{\alpha}$$ ...
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56 views

What is the physical significance of the fermi-field asymmetric stress-energy tensor?

Using the ideas from a previous question here it can be shown that if one takes the boson spin 1 stress-energy tensor of the form \begin{align} T^{\mu\nu}_{\text{spin one}} = \begin{bmatrix} ...
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56 views

Non-trivial components of the stress-energy tensor of the bosonic string ghost action

The stress-energy tensor derived from the ghost action of a bosonic string is: $$ T_{\alpha \beta} = \frac{i}{4 \pi} \left ( b_{\alpha \gamma} \nabla_{\beta} c^{\gamma} + b_{\beta \gamma} ...
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147 views

Perturbed stress-energy tensor in a cosmological context?

In the theory of cosmological pertubations, we can write the metric of a null-curvature expanding Universe as : $ds^2 = -c^2\left(1+2\frac{\psi}{c^2}\right)dt^2 + a^2 ...
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Warping function for torsion of non-circular prism

I have a few questions regarding the case of torsion of a prism, as encountered in continuum mechanics. Specifically, a prism (which can be a cylinder, a rectangular prism, elliptical prism, etc.) has ...
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16 views

An expression for stress power

I have seen it written that for a continuum undergoing deformation, if we ignore body forces and heat transfer, the work done is equal to stress power: $\cfrac{dW}{dt}=\sigma_{ij}D_{ij}$, where ...
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66 views

What is the explicit form of $\tau^{\alpha\beta}$ in the linearized Einstein field equations $\Box h^{\alpha\beta}=-16\pi\tau^{\alpha\beta}$?

If we let $h^{\alpha\beta}=\eta^{\alpha\beta}-g^{\alpha\beta}\sqrt{|det(g)|}$ then, according to wikipedia, the Einstein Field Equations become $$\Box h^{\alpha\beta}=-16\pi\tau^{\alpha\beta},$$ where ...
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38 views

Photon Gas Stress-Energy Tensor

In standard texts it is typically discussed how one can obtain the stress energy tensor of a perfect fluid, in both coordinate-dependent and coordinate independent forms: \begin{equation} ...
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45 views

How is momentum conserved in this electromagnetic scattering?

It is well known that verifying momentum and energy conservation in the presence of electromagnetic fields requires care as the fields themselves carry energy and momentum (see Griffiths chapter 8 for ...
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34 views

Breakdown of correspondence principle: null dust case

In both quantum and general relativity theories we are used to provide results in the "limited" conditions to demonstrate a correspondence between new and old formalism. For instance deflection of ...
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41 views

Stress-Energy Tensor of Riemann's Curvature Field

Einstein, in "The Foundation of the General Theory of Relativity," as well as most modern lecturers in the subject, use a pseudo-tensor for the stress-energy of the gravitational field. By ...
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How to obtain momentum four vector components from the stress energy tensor

In obtaining a momentum component as an integral of the relevant stress energy tensor component, I sometimes come across it shown as an integral of the tensor component over three space and sometimes ...
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33 views

Solving the Stress-energy-momentum tensor with metric

Let's say I want to find an expression for $T_{00}$ in Einstein field equations given a particular metric. I need to find first $g_{00}$, which is not complicated to find, and $R_{00}$ which is ...
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42 views

Hilbert Stress Energy Tensor for fermions + EM field and Yang-Mills theory (fermions + gluons)

@Qmechanic or anyone else who knows the reference. I am trying to find a references to the work(s) where thorough derivation of Hilbert Stress Energy Tensor for fermions + EM fields and Yang-Mills ...
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Show that $M^{\mu\nu}$ describes the angular momentum of the system

Define $M^{\mu\nu}$ = $\int d^3x(x^\mu T^{0 \nu}-x^{\nu}T^{0 \mu})$ describes the angular momentum of the system. I don't want you to solve it but I'm not really sure what kind of criterion it ...
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Prove that $T_{00}$, $T_{10}$, $T_{01}$, and $T_{11}$ are all $L/(4\pi x^2)$ at $(ct, x, 0, 0)$ for star of constant luminosity $L$

We have a star of constant luminosity $L$. We want to prove that the components $T_{00}$, $T_{10}$, $T_{01}$ and $T_{11}$ are all the same for the event $(ct,x,0,0)$ and they are all $L/(4\pi x^2)$. ...
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40 views

Symmetry of the Gravitational Stress Energy pseudo tensor

Recently, I have been reading on the Gravitational Stress-Energy pseudo tensor. It says in Wikipedia that one of the conditions for a suitable GSE pseudo tensor is that it has to be symmetric about ...
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55 views

Entropy of the boundary and stress-energy tensor in the bulk

The importance of this result cannot be understated: Positivity, monotonicity and convexity of relative entropy in the boundary is implied by the positivity of the stress-energy density tensor in the ...
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41 views

electroweak field contribution to the space curvature in GR

i've just found out that EM stress energy tensor along with gravitational stress energy contribute to the curvature of space. So, does the electroweak field also contribute to the curvature of space? ...
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162 views

Is there a general stress-energy tensor for vector fields?

I've been reading about scalar fields in the context of general relativity, and I found this page: https://en.wikipedia.org/wiki/Stress-energy_tensor#Scalar_field. It says that the stress-energy ...
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32 views

How does the photon field operator change are understanding of the electromagnetic stress-energy tensor?

Given the following photon field operators \begin{align} \mathbf{A}(\mathbf{r}) &=& \sum_{\mathbf{k},\mu} \sqrt{\frac{\hbar}{2 \omega V\epsilon_0}} \left(\mathbf{e}^{(\mu)} ...