What is Stress-energy tensor: Definition and 98 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. G

    Stress-energy tensor & mass term in metric

    I'm trying to clarify for myself the relation between the stress-energy tensor and the mass scalar term in metric solutions to Einstein's equations. Maybe I should also say I'm trying to understand the energy tensor better, or how it relates to boundary conditions on the solutions. My...
  2. M

    Stress-Energy Tensor from Lagrangian: Technical Question II

    This thread is supposed to be a continuation of the discussion of this thread: (1) https://www.physicsforums.com/showthread.php?t=88570. The previous thread was closed but there was a lot of things I did not understand. This is also somewhat related to a recent thread I created: (2)...
  3. M

    Stress-energy tensor for electromagnetic field with interaction term

    First of all, I'm not sure if this thread belongs here or at the "Special & General Relativity" sub-forum, if I posted at the wrong place please move it. Homework Statement I encountered this problem working in my master's degree. I need to find the stress-energy tensor of the following...
  4. N

    How to construct stress-energy tensor for a system?

    Given a particular system, how would one construct the stress-energy tensor? I was reading Mallett's paper and the stress-energy given for an infinitely long circulating cylinder of light is of the form T_{\mu\nu}=\epsilon \eta_\mu \eta_\nu where \eta_\mu=(\eta_0,0,\eta_2,0) and ε is the energy...
  5. fluidistic

    Density of energy from the stress-energy tensor

    Homework Statement Hi guys, I would like to show that if ##t^\mu## is a temporal vector then ##t^\mu t^\nu T_{\mu\nu}## is the density of energy of the EM field measured by an observer with velocity ##t^\mu##. And that it is greater or equal to 0. Density of energy is proportional to...
  6. fluidistic

    Trace of the stress-energy tensor

    0. Homework Statement Hi guys, I must show that the trace of the stress energy tensor is zero. The definition of it is ##T^{\mu \nu }=\frac{1}{4\pi} \left ( F^{\mu \sigma } F^{\nu \rho} \eta _{\sigma \rho}-\frac{1}{4} \eta ^{\mu \nu } F^{\sigma \rho} F_{\sigma \rho} \right )##. 1. The...
  7. C

    Construct electromagnetic stress-energy tensor for a non-flat metric

    Hi, I am having problems in constructing a stress-energy tensor representing a constant magnetic field Bz in the \hat{z} direction. The coordinate system is a cylindric {t,r,z,\varphi}. The metric signature is (+,-,-,-). I ended with the following mixed stress-energy tensor: Is this...
  8. C

    Stress-energy tensor definition

    I have seen two definitions with oposite signs (for one of the pressure terms in the formula) all over the web and books. I suspect it is related to the chosen metric signature, but I found no references to that. General Relativity An Introduction for Physicists from M. P. HOBSON...
  9. T

    Gauge invariance of stress-energy tensor for EM field

    For free EM field: L=-\frac{1}{4}FabFab Then the stress-energy tensor is given by: Tmn=-Fml∂vAl+\frac{1}{4}gmnFabFab The author then redefines Tmn - he adds ∂lΩlmn to it, where Ωlmn=-Ωmln. The redefined tensor is: Tmn=-FmlFvl+gmv\frac{1}{4}FabFab It is gauge invariant and still satisfies...
  10. ShayanJ

    The relationship between Stress-Energy tensor and Mass

    In Einstein field equations,the term that is responsible for curving Space-Time is the Stress-Energy tensor.But we know that mass should be able to curve space-time.So I think every mass distribution should have a Stress-Energy tensor associated with it. What is that relationship? Thanks
  11. Q

    Is There Really a Strictly Conserved Stress-Energy Tensor in GR?

    This is a fork off the locked thread here: https://www.physicsforums.com/showthread.php?t=648423, and is further a response to a recent blog entry 'Does Gravity Gravitate?' (not sure of the PF rules on blogs re threads so won't post a link to it here). The blog presents well what is doubtless a...
  12. bcrowell

    Asymmetric stress tensor gives asymmetric stress-energy tensor?

    I'm sure there's a trivial explanation for this, but it's escaping me. The space-space components of the stress-energy tensor are interpreted as the 3x3 stress tensor. But WP claims that the symmetry of the stress tensor need only hold in the case of equilibrium: "However, in the presence...
  13. A

    Stress-Energy Tensor for Gravitational Field

    I'm trying to understand the way that the stress-energy tensor for a gravitational field is derived and I've run into a few problems. It seems that there are two main avenues which are kind of similar. One derivation involves looking at g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu} where \eta_{\mu\nu} is...
  14. stevendaryl

    Spin Density and Non-symmetric Stress-Energy Tensor

    In General Relativity, the assumption is made that the stress-energy tensor Tαβ is symmetric. However, if there are particles with intrinsic spin, then this assumption is false, as described here: http://en.wikipedia.org/wiki/Spin_tensor The spin tensor Sαβμ satisfies: ∂μ Sαβμ = Tβα -...
  15. L

    Expectation value of Normal ordered Stress-energy tensor

    Hi, In Birrel and Davies ch4 they write: \langle \psi|:T_{ab}:|\psi \rangle =\langle \psi|T_{ab}|\psi \rangle -\langle 0|T_{ab}|0 \rangle this is for the usual Mink field modes and vac state. Why does normal ordering reduce to this expression, could anybody point me the way to...
  16. M

    Vector Potential Stress-Energy Tensor

    The vanishing divergence(s) of the stress-energy tensor, which proves/demands (not sure which) the conservation laws for mass-energy and momentum, would seem to suggest to a naive person (me) that there might be some sort of "vector potential" associated with the stress-energy tensor, similar to...
  17. R

    Why doesn't Noether thm produce exactly the stress-energy tensor?

    In classical field theory, use noether theorem to compute conserved currents for electromagnetic Lagrangian. \mathcal{L} = \frac{1}{4}F_{\mu\nu}F_{\mu\nu}, \quad F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu For arbitrary translational symmetries, the Noether conserved current evaluates...
  18. putongren

    Definition of Negative Pressure in the Stress-Energy Tensor

    I've been trying to find out what the definition of negative pressure in the context of General Relativity. There are some information on negative pressure and I found this: https://www.physicsforums.com/archive/index.php/t-299495.html The starter of that thread also doesn't know what the...
  19. S

    Covariant derivative of stress-energy tensor

    hi, I understand that Tab,b=0 because the change in density equals the negative divergence, but why do the christoffel symbols vanish for Tab;b=0?
  20. O

    What is the Stress-Energy Tensor

    I have been trying to self teach General Relativity through Wikipedia, mathematical "experiments," and Google, but no matter how much searching I do, I can't figure out what, exactly, the Stress-Energy Tensor is, or what the components mean.
  21. T

    Problems with a complex stress-energy tensor

    Hi , I am working with the following stress-energy tensor: T\mu\nu=\partial\mu\phi\partial\nu\phi* - g\mu\nu(\nabla\mu\phi\nabla\mu\phi* - m2\phi\phi*) Where \phi is a complex scalar of the form: \phi(r,t) = \psi(r)eiwt that obeys the Klein-gordon equation and \phi* is the complex...
  22. A

    Questions on Stress-Energy Tensor: Definition & 0 Value

    I am particularly new to tensors, so, just asking, could some give me the definition of a stress-energy tensor? Also, what would a stress-energy tensor of 0 indicate? Thanks in advance.
  23. A

    Stress-energy tensor in static cylindrical case

    I have some problems using this definition, maybe because it's not valid in every coordinate system: T^{\mu\nu} = (\epsilon + p) \frac{dx^{\mu}}{ds} \frac{dx^{\nu}}{ds} -p g^{\mu\nu} since in cylindrical coordinates x^0 =t \qquad x^1 =\rho \qquad x^2 = \phi \qquad x^3 =z using weyl metric...
  24. G

    Exploring the Possibility of an EM Stress-Energy Tensor with Equal Sigma-Values

    Is it possible for an EM stress-energy tensor such as this: [PLAIN]http://www3.telus.net/public/kots1906/emtensor.jpg to exist, where \sigma_{xx} = \sigma_{yy} ?
  25. michael879

    EM Stress-Energy Tensor Derivation: Understanding the Symmetry and Conditions

    Can someone please walk me through (or provide a link that does) the derivation of the EM stress-energy tensor? I get all the concepts I'm just a little confused on some of the details. Basically, you have the definition of the stress energy tensor in terms of the lagrangian, and the condition...
  26. P

    Interpreting Stress-Energy Tensor Components for Klein-Gordon & EM Fields

    I'm curious what is the interpretation of spatial components of stress-energy tensor in case of Klein-Gordon field or electromagnetic field. Time-time component is the energy density of the field, time-spatial components are the momentum density. The diagonal spatial components are probably...
  27. D

    Stress-energy tensor diagonalization

    This question probably applies to symmetric rank-2 tensors in general, but I've been thinking about it specifically in the context of the stress-energy tensor. For any stress-energy tensor and any metric (with signature -, +, +, +), is it possible to find a coordinate transformation that a)...
  28. A

    Worldsheet ghost stress-energy tensor

    Hey guys, I haven't posted on here for quite a while, so hello to everybody. I've been trying to derive the stress-energy tensor for the ghost LaGrangian: \int d^2 \sigma \sqrt{g} \left( b_{\alpha\beta} \nabla^\alpha c^{\beta} + \omega b^{\alpha\beta} g_{\alpha\beta} \right) for...
  29. G

    Elaborating on stress-energy tensor

    Hi, I'm trying to work out the logic behind a statement I found in the GR book I'm currently studying. It says that from the conservation equation \nabla_aT^{ab}=0, one could deduce the following two equations: (\varrho+p)\dot{u}^a = \nabla^ap - u^a\dot{p} \dot{\varrho} +...
  30. P

    Interpreting Stress-Energy Tensor in General Relativity

    What is the interpretation of stress energy-tensor in general relativity. According to General Relativity by Wald T^{\mu\nu} u_\mu u_\nu is energy density (chapter 4.2) where u_\mu is observer 4-veliocity. But for isolated particle T^{\mu\nu} = \gamma m V_\mu V_\nu...
  31. A

    Solving the Stress-Energy Tensor Problem

    Hi, How would go about arguing that the Stress-Energy tensor is actually a tensor based on how it must be linear in both it's arguments? I'm thinking it requires one 1-form to select the component of 4-momentum (e.g. \vec{E}=<\tilda{dt} ,\vec{P}> ) and also one 1-form to define the surface...
  32. maverick280857

    From Noether's Theorem to Stress-Energy Tensor

    Hi The following is a standard application of Noether's Theorem given in most books on QFT, in a preliminary section on classical field theory. Reproduced below are steps from the QFT book by Palash and Pal, which I am referring to, having read the same from other books. I have some trouble...
  33. A

    Rotating rigid sphere stress-energy tensor

    Hy. Can somebody please show me the way, how to transform stress-energy tensor for sphere in rest frame to stress-energy tensor in rotating frame using Lorentz transformations?
  34. M

    Stress-energy tensor proof (schutz ch7 q 8)

    Homework Statement Using the (previously proved) equation : if g_{\alpha \beta} is independent of x^\mu[/itex] then \frac{1}{\sqrt{-g}}(\sqrt{-g}{T^\nu}_\mu)_{,\nu} = 0 Prove that \int_{x_{0}=const} {T^\nu}_\mu (\sqrt{-g})n_\nu d^3 x is independent of x^0 if n_\nu is the unit...
  35. F

    Calculating Stress Energy Tensor for Rotating Bodies

    Hello, can someone explain to me how to calculate the stress energy tensor. If we remove the first row and column, we get get the stress tensor, which I am not familiar with, or am I missing something here? But what about the components of the first column and row. What I know so far is that...
  36. stevebd1

    Stress-energy tensor for a rotating object

    I'm currently looking at stress-energy tensors and while I understand that T00 is density and T11, T22 and T33 are pressure, where does angular momentum fit in this? I'm assuming the quantities T21, T31 and T32 might have something to do with this. For example, how would you incorporate the...
  37. stevebd1

    Stress-energy tensor and active mass

    Imagine a 2.2 sol mass neutron star on the brink of collapse with a radius of 12 km, an average density of 0.605e17 kg/m^3 and an average EOS of ~1/7. Based on active mass (i.e. including for pressure), the stress-energy tensor (g) would be based on g=\rho c^2+3P resulting in g≡3.143 sol. The...
  38. J

    Continuity equation from Stress-Energy tensor

    It is true that \frac{\partial}{\partial x^\beta} T^{0 \beta} = \gamma^2 c \left( \frac{\partial \rho}{\partial t} + \vec{\nabla} \bullet \left[ \rho \vec{v} \right] \right) = 0 but, how do we arrive at this point? What is in T^{ \alpha \beta} and how do we compute it for any...
  39. J

    Stress-energy tensor and pressure

    I learned that stress-energy tensor is defined in first place to be a 16-component object T^{\mu\nu}, where the first row T^{0\nu} tells the energy density current, and the three other rows T^{i\nu} tell the momentum density currents. The Carrol introduces a stress-energy tensor where off...
  40. P

    Stress-energy tensor of a wire under stress

    [Edit] switch to consistent geometric units and get rid of factors of 'c'. I've been thinking about the relativistic hoop/disk again, and as a preliminary step I decided to look at what happens to the stress-energy tensor of a wire when we apply a load to it. Suppose we have a wire, of mass m...
  41. C

    Interpreting the Einstein stress-energy tensor T_ab

    The Einstein field equation relates the curvature of space to the distribution of matter, representing the latter with a tensor T. The components of this tensor have been interpreted as representing the volume-density of mass-energy together with the "pressure" in each spatial direction. Can...
  42. N

    Stress-energy tensor of a perfect fluid

    The stress energy tensor of a perfect fluid is composed of two terms of which only one term contains the metric tensor gab. (product of metric tensor and pressure). For curved spacetime, one replaces the flat spacetime metric tensor by the metric tensor of curved space. What I find bizar however...
  43. N

    Stress-energy tensor in curved spacetime

    In any textbook on relativity, one finds the classical expression for the stress-energy tensor of a perfect fluid. In generalizing this tensor to curved spacetime, one just replaced the flat-spacetime metric tensor by the metric tensor of curved spacetime. It seems logical to do so, but in my...
  44. P

    Stress-energy tensor as a 2-form

    I've been looking off and on recently at how the stress-energy tensor can be interpreted as a 2-form. I can't quite see how one manages to convert a symmetric rank 2 tensor into a 2-form, though, given that a 2-form is by defintion anti-symmetric, in spite of some reading. I'm hoping that...
  45. P

    Stress-Energy tensor of a rotating disk

    I'm getting a rather crazy looking result, but I'm beginning to think it may be right. Unfortunately, I haven't been able to find any specific references on the topic to check my sanity level. Basically, I'm finding that in relativistic terms, there are no pressure (or tension) terms in the...
  46. T

    Does a black hole have a stress-energy tensor?

    Can the energy density of an empty but expanding spacetime decrease with expansion? Does a black hole have a stress-energy tensor?
  47. L

    Stress-Energy Tensor from Lagrangian: Technical Question

    Stress-Energy-Momentum Tensor from Lagrangian: Technical Question I've been reading about how to generate the stress-energy-momentum tensor T^{\mu \nu} from the action S = \int d^{4}x \sqrt{|g|} \mathcal{L} T^{\mu \nu} = \frac{2}{\sqrt{|g|}} \frac{\partial}{\partial g_{\mu \nu}} \left(...
  48. S

    I can't see how stress-energy tensor meets the minumum tensor requirement

    Gentlemen, I am sorry. I did a few typing errors here in order to put latex in and I even use 12 minute = 1 hour. This might confuse you. Let me try to correct this. Said , I use the simple dust model with 216,000 grams in a volume of 1 light-hour^3. So, T^\mu\nu = diag(216000 , 0, 0...
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