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.
My question is about this step in the derivation:
When the ##\partial_\nu \mathcal L## in 3.33 moves under the ##\partial_\mu## in 3.34 and gets contracted, I'd expect it to become ##\delta_{\mu \nu} \mathcal L##. Why is it rather ##g_{\mu \nu} \mathcal L## in the 3.34?
(In this text, ##g_{\mu...
This is just basic algebra for the energy-momentum relationship, but the calculations confuse me. May I ask what is wrong with my concept or calculation causing the following problem.
Maybe it's because I'm getting older, my ability to think and calculate has declined...
Hello,
this is my first thread.
Robert Wald, in General Relativity, equation (4.2.8) says :
E = – pa va
where E is the energy of a particle, pa the energy-momentum 4-vector and va the 4-velocity of the particle. How can I see this is compatible with the common energy-momentum-relation E2 – p2 =...
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'}})##
$$ \bra{ \vec{ p'}} T_{\mu,\nu} \ket{ \vec...
So I've managed to confuse myself on this problem :)
Since the problem says we can assume ##m_p << m_b##, I'm assuming that the velocity of the bowling ball will be unchanged, such that ##\vec v_{b,i} = \vec v_{b,f} = -v_{b,0} \hat i##
I started out using the energy-momentum principle, ##(\vec...
I know the tensor can be written as $$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}$$? And if so, is there a...
In a spacetime diagram the spatialized time direction is the vertical y-axis and the pure space direction is the horizontal x-axis, ct and x, respectively.
The faster you go and therefore the more kinetic energy you have, you'll have a greater component of your spacetime vector in the...
REMARK: First of all I have to say that this Lagrangian reminds me of the Lagrangian from which we can derive Maxwell's equations, which is (reference: Tong QFT lecture notes, equation 1.18; I have attached the PDF).
$$\mathcal{L} = -\frac 1 2 (\partial_{\mu} A_{\nu} )(\partial^{\mu} A^{\nu}) +...
Show that, according to relativistic physics, the final velocity ##v## of a rocket accelerated by its rocket motor in empty space is given by
##\frac{M_i}{M} = \Big ( \frac{c+v}{c-v} \Big) ^ \frac{c}{2 v_{ex}}##
where ##M_i## is the initial mass of the rocket at launch (including the fuel)...
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:
$$\begin{align*}
\delta S_M
&= \int d^4x (\delta\sqrt{-g})...
There are plentty of textbooks and online papers that talk about the energy momentum tensor, but they all look to me as if they're only covering the very introductory aspects of it. To put another way, it seems that there's much more to be learn.
I would like to know if university physics...
The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is
\begin{equation}
T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}.
\end{equation}
Let contract...
Photons deviate from the above energy-momentum transformations under certain circumstances while still in flat space-time, I'm wondering what set of transformations would more accurately describe them over as wide a range of circumstances as possible, still in flat space-time, I've searched and...
For a curve parametrised by ##\lambda## where ##\lambda## is along length of the curve and is 0 at one end point.
At each ##\lambda## say tangent vector V and A be the two possible vectors of the tangent space.
where ##V=V^\mu e_\mu## and ##A=A^\nu e_\nu##, {e} are the basis vectors.
Now ##...
Hi.
I'm reading an introductory text that somehow seems to confuse if ##E^2-(cp)^2=const## means that the left side is invariant (under Lorentz transformations) or conserved (doesn't change in time). As far as I understand it, they only prove Lorentz invariance.
Are they both true? If so...
Hi.
I've read that there's no Newtonian analogue of the energy-momentum relation
$$E^2-(pc)^2=(mc^2)^2\enspace .$$
Why doesn't
$$E=\frac{p^2}{2m}$$
qualify as such? There's no rest energy in Newtonian physics anyway.
In Special Relativity, we have the four vector, (E/c, px, py, pz). However, isn't the first term just `p` given that `E=pc` for a photon? Why is it an energy-momentum four vector when the first term isn't really energy but momentum?
Homework Statement
I want to be able, for an arbitrary Lagrangian density of some field, to derive the energy-momentum tensor using Noether's theorem for translational symmetry.
I want to apply this to a specific instance but I am unsure of the approach.
Homework Equations
for a field...
Homework Statement
This should be pretty simple and I guess I am doing something stupid?
##T_{bv}=(p+\rho)U_bU_v-\rho g_{bv}##
compute ##T^u_v##:
##T^0_0=\rho, T^i_i=-p##Homework Equations
##U^u=\delta^t_u##
##g_{uv}## is the FRW metric,in particular ##g_{tt}=1##
##g^{bu}T_{bv}=T^u_v##
##...
How do astrophysicists accurately account for all of the energy and pressure within a galaxy? How is it tabulated? My understanding of general relativity predicts that space-time curvature is a consequence of mass, energy, and pressure as expressed in the Energy-Momentum tensor.
The accepted...
Hi everyone,
Im a little bit confused about deBroglies procedure on introducing his famous Matterwave formula.
People already knew that the wavelength of the light was equal to Lambda = h/p. The term p comes from the energy-momentum formula; for the light the restmass = 0 so E =pc etc.
As...
Hi all - forgive me, I'd asked a series of questions in a previous post that was deemed to be circular, but I still didn't obtain a satisfactory answer to the question I was asking. In this post, I'm going to try to be very careful to use terms that are at least less 'misplaced', per se...
As you may know from some other thread, I was interested through the week in finding a general way of express the energy-momentum tensor that appears in one side of the Einstein's equation.
After much trials, I found that
$$T^{\sigma \nu} = g^{\sigma \nu} \frac{\partial \mathcal{L}}{\partial...
Hello,
In deriving the energy-momentum equation:
E^2 = (pc)^2 + (mc^2)^2
the following equations are used:
p = ymv
E = ymc^2
But both equations are equations that depend on mass, while the final result does not and applies to massless particles. Besides the energy-momentum equation is...
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:
$$ ds^2= -c^2dt^2 + a^2(t) \left( {\frac{dr^2}{1-kr^2} + r^2 d\theta^2 + r^2...
Dear all,
in a lot of undergraduate textbooks you find the claim that antiparticles can be motivated by Einstein's energy-momentum relation ## E^2 = p^2 + m^2 ##, which has both 'negative' and 'positive energy' solutions. In the context of a single wave function this is problematic. In the...
I'm trying to show that \partial_\mu T^{\mu \nu}=0 for
T^{\mu \nu}=F^{\mu \lambda}F^\nu_{\; \lambda} - \frac{1}{4} \eta^{\mu \nu} F^{\lambda \sigma}F_{\lambda \sigma},
with the help of the electromagnetic equations of motion (no currents):
\partial_\mu F^{\mu \nu}=0,
\partial_\mu F_{\nu...
As I understand it one is forced to use 4-vectors since we require objects that transform as vectors under application of Lorentz transformations and 3-vectors do not (technically they do under rotations, but not under boosts). Equivalenty, if one starts off with Minkowski spacetime from the...
Homework Statement
Maxwell's Lagrangian for the electromagnetic field is ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}## where ##F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}## and ##A_{\mu}## is the ##4##-vector potential. Show that ##\mathcal{L}## is invariant under gauge...
Homework Statement
The energy-momentum tensor ##T^{\mu\nu}## of the Klein-Gordon Lagrangian ##\mathcal{L}_{KG} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}## is given by
$$T^{\mu\nu}~=~\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}.$$
Show...
I apologize in advance for this dumb question, I think I know the answer but I just want to be sure.A photon has energy E = pc = hf
Do the Energy-Momentum transformations:
apply exactly to photons?
Or must we introduce certain corrective terms? Let's say all this takes place in free space.
Hi all,
I am trying to follow the calculation by samalkhaiat in this thread: https://www.physicsforums.com/threads/finding-equations-of-motion-from-the-stress-energy-tensor.547502/page-2 (post number 36). I am having some difficulty getting the equation above equation (11) (it was an unnumbered...
The Einstein field equation is inconsistent unless we demand a divergence-free stress-energy tensor. This makes me think that Hoyle's steady-state cosmology is inconsistent with general relativity.
But Hawking and Ellis has this at p. 90:
I had always imagined that the C field was just some...
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
[/B]
A particle of mass m is moving in the +x-direction with speed u and has momentum p and energy E in the frame S.
(a) If S' is moving at speed v, find the momentum p' and energy E' in the S' frame.
(b) Note that E' \neq E and p' \neq p, but show that...
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...
Hello,
##E_{tot}^2=(pc)^2+(m_0 c^2)^2## works fine for mass ##m_0## moving with relativistic speeds. What if the moving mass has internal energy also (say, heat). Does the energy-momentum relation still apply? What is the expression for the momentum ##p## then?
Because ##p=\gamma m_0 v##...
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?
Is this graphic wrong, see,
http://en.wikipedia.org/wiki/File:Spontaneous_Parametric_Downconversion.png
Shouldn't k_s + k_i be less than k_pump in the top graphic because |k_s| + |k_i| = |k_pump|, as energy is proportional to momentum?
If so is momentum transferred to the crystal after the...
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...