What is Stress energy tensor: Definition and 63 Discussions

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

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  1. Bishal Banjara

    I Obtaining the matter Lagrangian from the stress energy tensor

    Basically, the stress energy tensor is given by $$T_{uv}=-2\frac{\partial (L\sqrt{-g})}{\partial g^{uv}}\frac{1}{\sqrt{-g}}.$$ It makes easy to calculate stress energy tensor if the variation of Lagrangian with the metric tensor is known. But it is possible to retrieve matter Lagrangian if the...
  2. milkism

    Maxwell Stress components of the energy-stress-momentum tensor

    Question: Solution: I need help with the last part. I think my numerical factors are incorrect, even if I add the last term it will get worse. What have I done wrong, or is there a better way to deal with this?
  3. StenEdeback

    I do not understand stress energy tensor for fluids

    I do private studies on my own for fun and right now I read about relativistic field theory as a preparation for later studies of quantum field theory. I simply do not understand where equation 13.78 in Goldstein's "Classical Mechanics" third edition comes from. Please explain. Please also...
  4. T

    A Covariant Derivative of Stress Energy Tensor of Scalar Field on Shell

    Hi all, I am currently trying to prove formula 21 from the attached paper. My work is as follows: If anyone can point out where I went wrong I would greatly appreciate it! Thanks.
  5. Sciencemaster

    I Piecewise Functions in the Einstein Field Equations

    Let's say I want to describe a massive box in spacetime as described by the Einstein Field Equations. If one were to construct a metric in cartesian coordinates from the Minkowski metric, would it be reasonable to use a piecewise Stress-Energy Tensor to find our metric? (For example, having...
  6. D

    I Contracting the stress energy tensor

    I am trying to understand the scalar form of the Einstein field equations. I know that you can contract the stress-energy tensor using the metric. And for a perfect fluid model, this turns out to be the energy density summed with the pressure. This also gives the Ricci scalar. However, you can...
  7. ohwilleke

    B Special Cases of Stress-Energy Tensor in GR

    Background and Motivation The stress energy tensor of general relativity, as conventionally defined, has sixteen components. One of those component, conventionally component T00, also called ρ, is mass-energy density, including the E=mc2 conversion for electromagnetic fields. The other...
  8. M

    Conservation law for FRW metric

    My attempt: Realize we can work in whatever coordinate system we want, therefore we might as well work in the rest frame of the fluid. In this case ##u^a=(c,\vec{0})##. The conservation law reads ##\nabla^a T_{ab}=0##. Let us pick the Levi-Civita connection so that we don't have to worry about...
  9. D

    I Trace of the stress energy tensor

    The stress energy tensor has many forms based on the type of matter you are describing, dust, fluid, perfect fluid... is it true that the trace of all of these matter situations is invariant?
  10. P

    A Einstein Tensor and Stress Energy Tensor of Scalar Field

    Hi All. Given that we may write And that the Stress-Energy Tensor of a Scalar Field may be written as; These two Equations seem to have a similar form. Is this what would be expected or is it just coincidence? Thanks in advance
  11. Dale

    I Invariants of the stress energy tensor

    Does anyone know of a set of invariants for the stress energy tensor? In particular, I would like to know if there is a small set of linearly independent invariants, each of which (or at least some of which) have a clear physical meaning.
  12. J

    A Electromagnetic Stress Energy Tensor Formula (-,+,+,+)

    I am trying to find the correct formula for the electromagnetic stress energy tensor with the sign convention of (-, +, +, +). Is it (from Ben Cromwell at Fullerton College): $$T^{\mu \nu} = \frac{1}{\mu_0}(F^{\mu \alpha}F^{\nu}{}_{\alpha} - \frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha...
  13. H

    I Understanding the stress-energy tensor

    I have trouble understanding some terms in the stress-energy-tensor. For instance T^(12) stands for the flux of the x-component of momentum in the y-direction. But what does it means for the x-component of momentum to flow in the y direction? Since momentum is a vector should't the x-component...
  14. F

    I Get Relation from Stress-Energy Tensor Def.

    Starting from the following definition of stress-energy tensor for a perfect fluid in special relativity : $${\displaystyle T^{\mu \nu }=\left(\rho+{\frac {p}{c^{2}}}\right)\,v^{\mu }v^{\nu }-p\,\eta ^{\mu \nu }\,}\quad(1)$$ with ##v^{\nu}=\dfrac{\text{d}x^{\nu}}{\text{d}\tau}## and...
  15. B

    Stress energy tensor transformation

    Homework Statement Show that if you add a total derivative to the Lagrangian density ##L \to L + \partial_\mu X^\mu##, the energy momentum tensor changes as ##T^{\mu\nu} \to T^{\mu\nu}+\partial_\alpha B^{\alpha\mu\nu}## with ##B^{\alpha\mu\nu}=-B^{\mu\alpha\nu}##. Homework EquationsThe Attempt...
  16. BiGyElLoWhAt

    I How to fill the stress energy tensor for multi body systems

    Say I wanted to set up EFE for the Earth and moon. How do I actually go about filling the stress energy tensor? I'm referencing the wikipedia page. So the time-time should be approximately E/c^2V, so for the Earth moon system ##T_{00} = \frac{3}{4\pi r_E^3}\frac{1}{c^2}(M_Ec^2 + 2/5...
  17. S

    I Einstein Field Eqs: Stress Energy Tensor Explained

    Hello! I have just started the Einstein field equations in my readings on GR and I want to make sure I understand the stress energy tensor. If we have a spherical, non-moving, non-spinning source, let's say a neutron star (I don't know much about neutron stars, so I apologize if the non-moving...
  18. S

    Transforming Stress-Energy Tensors in Different Frames

    Homework Statement In an inertial frame O calculate the components of the stress–energy tensors of the following systems: (a) A group of particles all moving with the same velocity ##v = \beta e_x##, as seen in O. Let the rest-mass density of these particles be ##\rho_0##, as measured in...
  19. S

    I Stress Energy Tensor: How Does It Implicate Flow of Energy?

    Hello! I am reading that in a perfect fluid we have no heat conduction, which implies that energy can flow out of a fluid element only if particles flow, so ##T^{0i} = 0##. I am not sure I understand why. We have ##\Delta E = \Delta Q - p \Delta V##. In our case as Q is constant, ##\Delta E = -p...
  20. S

    I Understanding Stress Energy Tensor of Fluids

    Hello! I am reading about stress energy tensor of a perfect fluid and I don't understand the ##T^{ij}## terms. They are defined to be the flux of i-th momentum through the j-th surface. Now you take a fluid element and in its momentary comoving reference frame (MCRF) you calculate these...
  21. S

    Is the speed of light always constant in nature?

    When I read the speed of light in vacuum is c, does it imply that light doesn't actually travel at this speed in nature? My guess is no, light always travels at c, it's just that in the definition, we're trying to ignore the affects on light from other stuff like the comological constant and the...
  22. K

    I Calculating Gravitational Mass of Compressed Gas Cylinder

    How do I calculate the gravitational mass of a cylinder of compressed gas, including the effects of pressure? By gravitational mass, I mean what I would measure on an ideal mass balance. (I know that the pressure is negligibly small in a realistic container, but I want to have a conceptual...
  23. Devin

    I Stress Energy Tensor for Oscillator: Setup for Electron in E-Field

    How would one go about setting up the stress energy tensor for a particle, say an electron subjected to electric an electric field that makes the particle oscillate with frequency \omega?
  24. A

    I Derivation of E.M. Stress Energy Tensor

    From Carroll (2004) It is possible to derive the Einstein Equations (with ##c=1##) via functional variation of an action $$S=\dfrac{S_H}{16\pi G}+S_M$$ where $$S_H= \int \sqrt{-g}R_{\mu\nu}g^{\mu\nu}d^4 x$$ and ##S_M## is a corresponding action representing matter. We can decompose ##\delta...
  25. mertcan

    A Stress energy tensor general relativity

    Hi, I would like say that in this link ( ) and starting from 56.28 Suskind tries to find the energy tensor equation using \phi, afterwards he finds a equation similar to wave equation in terms of \phi. My question is: For what does \phi stand ? I could not capture the meaning of \phi. Could...
  26. R

    Stress Energy Components: 3 or 4 Densities?

    I usually come across expressions for the Stress energy tensor showing them as three densities (normally over a space-like slice. In wikipedia's article on the spin tensor, they clearly (it does not appear to be an error) both write about and express the components of the stress energy tensor as...
  27. PerpStudent

    Dimension of the stress energy tensor

    The coefficient of the stress energy tesor in the GR equation reduces to 8π/Ν, where N = {"(Kg)m/s^2.} Is it correct to conclude that all the elements of the stress energy tensor must have the dimension of N = (Kg)m/s^2 since the curvature and metric tensors on the other side of the equation are...
  28. U

    Faraday Tensor and Index Notation

    Homework Statement (a) Find faraday tensor in terms of ##\vec E## and ## \vec B ##. (b) Obtain two of maxwell equations using the field relation. Obtain the other two maxwell equations using 4-potentials. (c) Find top row of stress-energy tensor. Show how the b=0 component relates to j...
  29. S

    General Relativity: Curvature and Stress Energy Tensor

    Hello all, I have a quick question regarding the relation of the space-time metric and the curvature. I have determined the space-time metric, g_(alpha beta), but I am unsure as how to go from the line element ds^2 = [ 1 + (dz/dr)^2] dr^2 + r^2 dtheta^2 and the space-time metric g to the...
  30. D

    Four divergence of stress energy tensor

    Homework Statement Hi, I'm trying to show the four divergence of the stress energy tensor of the sourceless klein gordon equation is zero. I got to the part in the solution where I am left with the equations of motion which is identically zero and 3 other terms. I managed to find a solution...
  31. P

    Divergence of the Stress-Energy Tensor

    Im studying Quantum Field Theory as part of my undergraduate course, and am currently looking at Noether's Theorem which has led me to the following calculation of the divergence of the Stress-Energy Tensor. I'm having difficulty in seeing how we get from line (31) to line (32). Is the 2nd term...
  32. S

    Do all elements of the stress energy tensor really have the same units?

    Some people may remember awhile back when I made a thread showing how when I derived the Einstein tensor and the stress energy momentum tensor for a certain traversable wormhole metric, that the units of the energy momentum tensor were not the same for each element and how a couple of the...
  33. S

    Stress Energy Tensor Components

    I have pretty much learned how to derive the left side of Einstein's field equations now (the Einstein tensor that is). Now I need to grasp that stress energy momentum tensor. Does anybody know of any good sources that will tell me how to derive the components of this tensor? I ask this...
  34. E

    Why shear stress components of the Stress Energy tensor not zero?

    Hi, I am having trouble understanding why Tij can be non-zero for i≠j. Tij is the flux of the i-th component of momentum across a surface of constant xj. Isn't the i-th component of momentum tangent to the surface of constant xj and therefore its flux across that surface zero? What am I...
  35. M

    Why Does the Stress-Energy Tensor Conservation Lead to a Surface Integral?

    Homework Statement Hello I'm trying to self study A First Course in General Relativity (2E) by Schutz and I've come across a problem that I need some advice on. Here it is: Use the identity Tμ\nu,\nu=0 to prove the following results for a bounded system (ie. a system for which Tμ\nu=0...
  36. W

    Conserved Charges of Stress Energy Tensor

    Hello, Hi There I am trying to obtain the relations of the conserved charges of the stress tensor, it has 4, one is the hamiltonian and the other three are the momentum components. \vec{P}=-\int d^3y \sum_i{(-\pi_i(y) \nabla \phi_i(y))} And i have to prove the conmutators...
  37. P

    Stress energy tensor for fields

    In the case of swarms of particles, the stress energy tensor can be derived by considering the flow of energy and momentum "carried" by the particles along their worldlines. Is there a way to interpret the field definition of the stress energy tensor from Wald, p455 E.1.26 T_{ab} \propto...
  38. A

    Stress Energy Tensor - Confusion about signs and metrics

    Hi everyone. I'm working on deriving Friedmanns Equations from the Einstein Field Equations. I've got the '00' components worked out but I'm having some trouble with the spatial indices 'ii' of the stress energy tensor ## T_{ii} ##. I'm the FLRW metric with c=1 and signature (-,+,+,+) so that...
  39. B

    Perfect fluids and the stress energy tensor

    Homework Statement "Texbooks that describe perfect fluids are often a little unclear about what is being assumed. It may not be immediately obvious why can't the pressures be different in different directions? Let's examine this. Suppose Tαβ = diag(ρ,(1+ε)P,P,P) . Show that if one performs a...
  40. shounakbhatta

    Energy density in Stress Energy Tensor

    Hello, I am new to this, if somebody could help me: The energy density in stress-energy-tensor is the amount of energy stored in a given system per unit mass. The T00=energy density. In a 4x4 matrix, we generally start with 1,1. Why it is 0,0? Secondly, what does energy density mean...
  41. D

    Finding stress energy tensor of a rod under tension

    I am given a rod with mass per unit length \mu , cross sectional area A, and under a tension F. I am also told that the tension F is uniform over the cross sectional area A. Find the stress energy tensor inside the rod. I know that for the stress energy tensor T^{00} gives the energy...
  42. C

    Stress energy tensor in extended standard model

    Maybe this is the wrong topic for this forum, but i am reading the following http://arxiv.org/abs/hep-ph/9703464 Which is on CPT violation and the standard model. I do not understand how they get to equation (9), when they write down the stress energy tensor as \Theta\mu\nu=1/2 i...
  43. Matterwave

    Finding Equations of Motion from the Stress Energy Tensor

    So, I'm reading Wald, and in it he talks about how the divergence-free nature of the stress-energy tensor implies "a lot" of knowledge about how matter moves in a curved space time. I'm wondering, how much is "a lot"? Can we obtain the full equations of motion from this? Wald gives the example...
  44. Z

    Stress energy tensor for a swarm of particles ( in MTW )

    hey Folks, Please see attachment. I'm in doubt about the equation 5.15a. Indeed, it is said on the line just above it that : "in a frame where particles have velocity Va" which means the lab frame , say. Then in this frame the time component of 4-velocity is the Lorentz contraction factor...
  45. A

    Stress Energy Tensor in 5D Space-Time: Action for Brane and Hidden Brane

    In a space time 5D, the action for the brane 4D is: \int dx^4 \sqrt{-h} In the Randall Sundrum the action for the hidden brane is: V_0\int dx^4 \sqrt{-h}, where V_0 is the tension on the brane hidden. follow the stress energy tensor T_{MN}= V_0 h_{uv} \delta^u_M \delta^v_N...
  46. R

    Experimental tests of the stress energy tensor

    hi, I wonder if the stress energy tensor formulation of general relativity has been well experimentally verified, since the schwarzchild metric is used most of the time. for example, is there any experimental evidence that an increase of the pressure with the same energy density, increases...
  47. R

    Electromagnetic stress energy tensor

    hi, I would like to know if the contribution of some electromagnetic field through the electromagnetic stress energy tensor can decrease the intensity of gravity. it has negative components. can it decrease pressure? in such case, this would lead to some kind of relative antigravity...
  48. J

    Help with differential geometry: Hilbert's stress energy tensor

    When varying the Hilbert action, we define the stress-energy tensor as: T_{\mu\nu}:= \frac{-2}{\sqrt{-g}}\frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{matter})}{\delta g^{\mu\nu}} = -2 \frac{\delta \mathcal{L}_\mathrm{matter}}{\delta g^{\mu\nu}} + g_{\mu\nu} \mathcal{L}_\mathrm{matter} I am...
  49. P

    Special relativity the EM stress energy tensor

    Homework Statement Using the expression below for the stress energy tensor of the em field, show that it has zero trace. Homework Equations T^{\mu\nu}=F^{\mu}_{ \alpha}F^{\alpha\nu}+\frac{1}{4}\eta^{\mu\nu}F_{\beta\gamma}F^{\beta\gamma} F is the faraday tensor and eta is the...
  50. R

    How does the Stress Energy tensor relate to Noether's theorem?

    Hi, I was wondering if the stress-energy tensor arose naturally in special relativity in the same way that plain energy and momentum do via Lagrangians. I understand Noether's theorem for particles, but Wikipedia describes the stress-energy tensor as a Noether current; can anyone explain what...