What is Energy-momentum: Definition and 103 Discussions

In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It is the extension of mass–energy equivalence for bodies or systems with non-zero momentum. It can be written as the following equation:

This equation holds for a body or system, such as one or more particles, with total energy E, invariant mass m0, and momentum of magnitude p; the constant c is the speed of light. It assumes the special relativity case of flat spacetime. Total energy is the sum of rest energy and kinetic energy, while invariant mass is mass measured in a center-of-momentum frame.
For bodies or systems with zero momentum, it simplifies to the mass–energy equation



E
=

m

0



c

2




{\displaystyle E=m_{0}c^{2}}
, where total energy in this case is equal to rest energy (also written as E0).
The Dirac sea model, which was used to predict the existence of antimatter, is closely related to the energy–momentum relation.

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  1. J

    Solving energy-momentum equations for lamba decay

    Homework Statement A lambda particle decays into a proton (at rest) and a pion. The rest masses are: lambda: 1116 MeV/c^2 pion: 140 MeV/c^2 proton: 938 MeV/c^2 we want to find the energy of the a) pion b) lambda (before decay) Homework Equations I am assuming we need to use the...
  2. W

    Gravitational energy-momentum tensor

    why general relativity can't define any tensorial expression for Gravitational energy momentum density ?
  3. N

    Energy-momentum tensor for electromagnetism

    Homework Statement Derive Tμν=FμλFνλ-1/4ημνFλθFλθ From \mathcal{L}=1/4F_{μν}F^{μν}+A_μJ^μ Homework Equations Above 3. The Attempt at a Solution The first term of the given equation and the second term of the equation to prove are i believe the same.i know, Jμ=\partial_νF^{μν}...
  4. A

    The Energy-Momentum Tensor

    I am a bit confused here. In the Einstein Field Equation, there is a tensor called stress-energy tensor in wikipedia and energy-momentum tensor in some books or papers which is $$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta(\mathcal{L} \sqrt{-g})}{\delta g_{\mu\nu}}$$ Is it equivalent to the...
  5. A

    Deriving the Metric from the Energy-Momentum Tensor

    Say we were given an expression for the energy-momentum tensor (also assuming a perfect fluid), without getting into an expression with multiple derivatives of the metric, are there any cases where it would be possible to deduce the form of the metric?
  6. P

    Geodesic Equation from conservation of energy-momentum

    Hi everyone, While reading http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html reference I bumped into a result. Can anyone get from Eq.19.1 to Eq.19.3? I've also been struggling to get from that equation to the one before 19.4 (which isn't numbered)...anyone? Thank...
  7. P

    Energy-Momentum Tensor for a particle

    Hello everyone, I was studying how to define, formally, an energy-momentum tensor for a point particle. I was reading this two references:http://academic.reed.edu/physics/courses/Physics411/html/page2/files/Lecture.19.pdf , page 1; and http://th-www.if.uj.edu.pl/acta/vol29/pdf/v29p1033.pdf...
  8. K

    Proving Homogeneous & Isotropic FRW Universe Energy-Momentum Tensor

    Hi everyone, It's not a real homework problem, but something I am trying to do that I haven't found in the literature. I am still stating the problem as if it was a homework Homework Statement Consider a FRW Universe. That is, ℝ x M, where M is a maximally symmetric 3-manifold, with a RW...
  9. A

    Energy-momentum pseudotensor example problem

    Homework Statement I'm following the derivation of finding the energy flux of a gravitational wave propagating along the z-axis where they use the energy-momentum pseudotensor to achieve this, but I can't seem to get an answer that matches theirs. Homework Equations We are given a general...
  10. N

    Graviton Propagator and energy-momentum tensor

    Dear PF, I am a little bit confused could you pls help me ... Suppose I a have a scatering or conversion of two particles via graviton propagator. Graviton propagator couples with energy-momentum tensor of matter fields. So can i assume that vertex to which graviton propagator is coupled...
  11. T

    Energy-momentum tensor for the Dirac spinor

    Hi there, I'm having a problem calculating the energy momentum tensor for the dirac spinor \psi (x) =\left(\begin{align}\psi_{L1}\\ \psi_{L2}\\\psi_{R1}\\ \psi_{R2}\end{align}\right)(free theory). So, with the dirac lagrangian \mathcal{L}=i\bar{\psi}\gamma^\mu\partial_\mu\psi-m\bar{\psi}\psiin...
  12. R

    Deriving the Energy-Momentum Formula

    Homework Statement Show that the energy-momentum relationship, E^2 = p^2 * c^2 + (m*c^2)^2, follows from the expressions E = (gamma)*m*c and p = (gamma)*m*u where (gamma) = 1 / sqrt(1 - (u^2)/(c^2)) the lorentz transformation factor. m is the rest mass. c is the speed of light u is the...
  13. P

    Deriving the Navier-Stokes equation from energy-momentum tensor

    The energy-momentum tensor for a perfect fluid is T^{ab}=(\rho_0+p)u^au^b-pg^{ab} (using the +--- Minkowski metric). Using the conservation law \partial_bT^{ab}=0, I'm coming up with (\rho+\gamma^2p) [\frac{\partial\mathbb{u}}{{\partial}t}+ (\mathbb{u}\cdot\mathbb{\nabla})\mathbb{u}]=...
  14. T

    Calculating the energy-momentum tensor for Maxwell Lagrangian

    Hi guys, can you help me with this? I'm supposed to calculate the energy momentum for the classic Maxwell Lagrangian, \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} , where F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu with the well known formula: T^{\sigma\rho}=\frac{\delta\mathcal{L}}{\delta...
  15. P

    Relativity energy-momentum tensor

    Homework Statement Arrive at the orthogonality relation {T^{\mu}}_{\alpha}{T^{\alpha}}_{\nu} = K{\delta^{\mu}}_{\nu} and determine K. Homework Equations T_{ij}=T_ji} The Attempt at a Solution {T^{\mu}}_{\alpha}{T^{\alpha}}_{\nu} = {T^{\mu}}_0{T^0}_{\nu}+ {T^{\mu}}_i{T^i}_{\nu} I am not...
  16. C

    Divergence of Energy-momentum Tensor

    How do you prove that Maxwell's energy-momentum equation is divergence-free? I don't know whether or not I have to use Lagrangians or Eistein's tensor, or if there's a simlpler way of expanding out the tensor.. ∂_{\mu}T^{\mu\nu}=0...
  17. C

    Divergence of Energy-momentum Tensor

    How do you prove that the energy-momentum tensor is divergence-free? ∂μTμν=0
  18. P

    Value of energy-momentum tensor in weak field approximations

    My first question, so sorry if it's in the wrong forum. I'm trying to understand the Newtonian weak field approximations to general relativity. I can't see why, if the Schwarzschild metric (which can describe the gravitational field around the Sun) is a vacuum solution (T_{\mu\nu}=0 ) , do...
  19. J

    What is the electrodynamic action and its energy-momentum tensor?

    I have studied Jackson, Landau, and Barut textbooks on electrodynamics, together with Weinberg's Gravitation and Cosmology textbook, and I find that the usual action S = S_f + S_m + S_{mf} is inconsistent and not well-defined. For instance, what is the meaning of S_f? A free-field term? Or...
  20. F

    Energy-momentum of gravitational waves

    Hello, I was wondering, since gravitational waves carry energy-momentum, would it be possible to find them in regions where the components of the metric tensor vanish? That is to say, empty space (non-quantum) is described by a vanishing energy-momentum tensor - but then, if gravitational waves...
  21. atyy

    Energy-momentum of non-free classical particle

    Let's discuss only classical fields and particles. For fields, E2=p2+m2 applies only if the field is free. In the presence of sources, we have to use the energy-momentum tensor. For particles, does E2=p2+m2 apply only when they are free, or does it work even if they are acted on by a force?
  22. A

    Energy-momentum tensor: metric tensor or kronecker tensor appearing?

    Hi This might be a stupid question, so I hope you are patient with me. When I look for the definition of the energy-momentum tensor in terms of the Lagrangian density, I find two different (?) definitions: {T^\mu}_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial_\nu...
  23. bcrowell

    SR pedagogy: energy-momentum and area in the x-t plane

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  24. T

    General Tensor contraction: Trace of Energy-Momentum Tensor (Einstein metric)

    Okay so I have: Eqn1) Tij=\rhouiuj-phij = \rhouiuj-p(gij-uiuj) Where Tij is the energy-momentum tensor, being approximated as a fluid with \rho as the energy density and p as the pressure in the medium. My problem: Eqn2) Trace(T) = Tii = gijTij = \rho-3p My attempt: Tr(T) = Tii...
  25. L

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  26. SamRoss

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  27. Q

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  28. K

    Origin of the Maxwell energy-momentum tensor?

    Electrodynamics force is f_i=F_{ik}j^k=F_{ik}\partial_j F^{jk}. I claim that the only way to obtain the Maxwell energy-momentum tensor T_i^j=-F_{ik}F^{jk}+\delta_i^jF_{kl}F^{kl}/4 is to write the force as a divergence: f_i=-\partial_jT_i^j.
  29. B

    Why is energy-momentum tensor Lorentz invariant?

    I'm studying General Relativity and facing several problems. We know that energy-momentum must be Lorentz invariant in locally inertial coordinates. I am not sure I understand this point clearly. What is the physics behind?
  30. Phrak

    Energy-Momentum Equation of a Particle

    How do I get from the Energy-Momentum equation of a particle to its Stress-Energy equation? By way of introducing the energy-momentum equation: For a single particle, in units where c=1, a relationship between mass, energy and momentum appear as a direct result of the 4-velocity: m^2 =...
  31. M

    Beta functions and the energy-momentum tensor

    Hi all, In Polchinski's string theory text he asserts (volume 1, section 3.4) that the trace of the energy-momentum tensor of a classically scale -invariant theory becomes proportional in the quantum theory to the beta function of the coupling, as a general point of QFT. This makes a kind of...
  32. D

    Is There More to the Energy-Momentum Equation Than Meets the Eye?

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  33. J

    Is the Higgs Vacuum Energy-Momentum Affected by Spontaneous Symmetry Breaking?

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  34. H

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  35. C

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  36. R

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  37. E

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  38. R

    Energy-momentum tensor and conservation of both energy and momentum

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  39. M

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  40. K

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  41. Y

    Modification of Energy-Momentum Relation and UV/IR mixing

    >From a seminar, I heard that energy-momentum relation (E^2=m^2+p^2) is modified by UV/IR mxing. In other words, the speaker claimed that the lowest energy is achived not by zero momentum, but by non-zero momentum. Could somebody refer me to a relevant paper? Thanks in advance Youngsub
  42. J

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  43. L

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  44. 1

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  45. E

    How Is the Classical Energy-Momentum Relation Derived?

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  46. I

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  47. I

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  48. J

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  49. C

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  50. J

    How to Calculate the Length of Energy-Momentum Four-Vectors?

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