What is Energy-momentum tensor: Definition and 59 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.
I've tried this problem so, so, so so so many times. Given the equations above, the proof starts easily enough:
$$\partial_\mu T^{\mu\nu}=\partial_\mu (∂^μ ϕ∂^ν ϕ)-\eta^{\mu\nu}\partial_\mu[\frac{1}{2}∂^2ϕ−\frac{1}{2}m^2ϕ^2]$$
apply product rule to all terms
$$=\partial^\nu \phi \cdot...
Question:
Solution:
I need help with the last part.
I think my numerical factors are incorrect, even if I add the last term it will get worse. What have I done wrong, or is there a better way to deal with this?
Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units?
Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR?
If so, What...
Can the energy-momentum tensor of matter and energy be cast in terms of energy density of matter and energy, similar to how the energy-momentum tensor of vacuum energy can be cast in terms of the energy density of vacuum energy?
I think it is quite simple as an exercise, following the two relevant equations, but at the beginning I find myself stuck in going to identify the lagrangian for a relativistic system of non-interacting particles.
For a free relativistic particle I know that lagrangian is...
Relevant Equations:: ##\ket{\vec{p}}=\hat{a}^{\dagger}(\vec{p})\ket{0}## for a free field with ##[\hat{a}({\vec{k})},\hat{a}^{\dagger}({\vec{k'})}]=2(2\pi)^3\omega_k\delta^3({\vec{k}-\vec{k'}})##
$$ \bra{ \vec{ p'}} T_{\mu,\nu} \ket{ \vec...
I know the tensor can be written as $$T^{\mu v}=\Pi^{\mu}\partial^v-g^{\mu v}\mathcal{L}$$ where $$g^{\mu v}$$ is the metric and $$\mathcal{L}$$ is the Lagrangian density, but how would I write $$T_{\mu v}$$? Would it simply be $$T_{\mu v}=g_{\mu \rho}g_{v p}T^{\rho p}$$? And if so, is there a...
REMARK: First of all I have to say that this Lagrangian reminds me of the Lagrangian from which we can derive Maxwell's equations, which is (reference: Tong QFT lecture notes, equation 1.18; I have attached the PDF).
$$\mathcal{L} = -\frac 1 2 (\partial_{\mu} A_{\nu} )(\partial^{\mu} A^{\nu}) +...
In Special Relativity I'm given the energy-momentum tensor for a perfect fluid:$$
T^{\mu\nu}=\left(\rho+p\right)U^\mu U^\nu+p\eta^{\mu\nu}
$$where ##\rho## is the energy density, ##p## is the pressure, ##U^\mu=\partial x^\mu/\partial\tau## is the four-velocity of the fluid. In the...
My attempt was to first rewrite ##S_M## slightly to make it more clear where ##g_{\mu\nu}## appears
$$S_M = \int d^4x \sqrt{-g} (g^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi-\frac{1}{2}m^2\phi^2).$$
Now we can apply the variation:
$$\begin{align*}
\delta S_M
&= \int d^4x (\delta\sqrt{-g})...
There are plentty of textbooks and online papers that talk about the energy momentum tensor, but they all look to me as if they're only covering the very introductory aspects of it. To put another way, it seems that there's much more to be learn.
I would like to know if university physics...
The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is
\begin{equation}
T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}.
\end{equation}
Let contract...
The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is
\begin{equation}
T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}.\tag{2}
\end{equation}
The covariant...
Homework Statement
I want to be able, for an arbitrary Lagrangian density of some field, to derive the energy-momentum tensor using Noether's theorem for translational symmetry.
I want to apply this to a specific instance but I am unsure of the approach.
Homework Equations
for a field...
Homework Statement
This should be pretty simple and I guess I am doing something stupid?
##T_{bv}=(p+\rho)U_bU_v-\rho g_{bv}##
compute ##T^u_v##:
##T^0_0=\rho, T^i_i=-p##Homework Equations
##U^u=\delta^t_u##
##g_{uv}## is the FRW metric,in particular ##g_{tt}=1##
##g^{bu}T_{bv}=T^u_v##
##...
How do astrophysicists accurately account for all of the energy and pressure within a galaxy? How is it tabulated? My understanding of general relativity predicts that space-time curvature is a consequence of mass, energy, and pressure as expressed in the Energy-Momentum tensor.
The accepted...
As you may know from some other thread, I was interested through the week in finding a general way of express the energy-momentum tensor that appears in one side of the Einstein's equation.
After much trials, I found that
$$T^{\sigma \nu} = g^{\sigma \nu} \frac{\partial \mathcal{L}}{\partial...
Hi everyone,
I want to derive the Friedmann equations from Einstein Field Equations. However, I have a problem that stems from the energy-momentum tensor. I am also trying to keep track of ## c^2 ## terms.
FRW Metric:
$$ ds^2= -c^2dt^2 + a^2(t) \left( {\frac{dr^2}{1-kr^2} + r^2 d\theta^2 + r^2...
I'm trying to show that \partial_\mu T^{\mu \nu}=0 for
T^{\mu \nu}=F^{\mu \lambda}F^\nu_{\; \lambda} - \frac{1}{4} \eta^{\mu \nu} F^{\lambda \sigma}F_{\lambda \sigma},
with the help of the electromagnetic equations of motion (no currents):
\partial_\mu F^{\mu \nu}=0,
\partial_\mu F_{\nu...
Homework Statement
Maxwell's Lagrangian for the electromagnetic field is ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}## where ##F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}## and ##A_{\mu}## is the ##4##-vector potential. Show that ##\mathcal{L}## is invariant under gauge...
Homework Statement
The energy-momentum tensor ##T^{\mu\nu}## of the Klein-Gordon Lagrangian ##\mathcal{L}_{KG} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}## is given by
$$T^{\mu\nu}~=~\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}.$$
Show...
I'm trying to show that \int d^3x \,x^\mu \left(\partial_\mu \partial_0-g_{\mu 0} \partial^2 \right)\phi^2(x)=0 . This term represents an addition to a component of the energy-momentum tensor \theta_{\mu 0} of a scalar field and I want to show that this does not change the dilation operator...
I'm looking at 'Lecture Notes on General Relativity, Sean M. Carroll, 1997'
Link here:http://arxiv.org/pdf/gr-qc/9712019.pdf
Page 221 (on the actual lecture notes not the pdf), where it generalizes that the energy-momentum tensor for radiation - massive particles with velocities tending to...
Homework Statement
(a) Show acceleration is perpendicular to velocity
(b)Show the following relations
(c) Show the continuity equation
(d) Show if P = 0 geodesics obey:
Homework EquationsThe Attempt at a SolutionPart (a)
U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v U^{\mu} +...
Hey guys,
So I have the stress energy tensor written as follows in my notes for the complex Klein-Gordon field:
T^{\mu\nu}=(\partial^{\mu}\phi)^{\dagger}(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial^{\nu}\phi^{\dagger})-\mathcal{L}g^{\mu\nu}
Then I have the next statement that T^{0i} is...
Homework Statement
I try to calculate the energy tensor, but i can't do it like the article, and i don't know, i have a photo but it don't look very good, sorry for my english, i have a problem with a sign in the result
Homework Equations
The Attempt at a Solution
In the photos...
I understand energy-momentum tensor with contravariant indices, where
I think I get T^{αβ}, but how do I derive the same result for T_{αβ}? Why are the contravariant vectors simply changed to covariant ones, and why does it work in Einstein's equation?
Homework Statement
Consider a stationary solution with stress-energy ##T_{ab}## in the context of linearized gravity. Choose a global inertial coordinate system for the flat metric ##\eta_{ab}## so that the "time direction" ##(\frac{\partial }{\partial t})^{a}## of this coordinate system agrees...
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}...
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...
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?
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...
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...
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...
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...
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}]=...
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...
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...
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...
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...
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...
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...
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...
Homework Statement
The problem is conveniently located here:
http://www.dur.ac.uk/resources/cpt/graduate/lectures/mscps.pdf
Problem no. 31. There's even a solution, here:
http://www.dur.ac.uk/resources/cpt/graduate/lectures/grsolns.pdf
However, I don't understand the solution...