What is Lorentz invariance: Definition and 101 Discussions
In relativistic physics, Lorentz symmetry, named after Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers that are moving with respect to one another within an inertial frame. It has also been described as "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".Lorentz covariance, a related concept, is a property of the underlying spacetime manifold. Lorentz covariance has two distinct, but closely related meanings:
A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, four-tensors, and spinors. In particular, a Lorentz covariant scalar (e.g., the space-time interval) remains the same under Lorentz transformations and is said to be a Lorentz invariant (i.e., they transform under the trivial representation).
An equation is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities (confusingly, some use the term invariant here). The key property of such equations is that if they hold in one inertial frame, then they hold in any inertial frame; this follows from the result that if all the components of a tensor vanish in one frame, they vanish in every frame. This condition is a requirement according to the principle of relativity; i.e., all non-gravitational laws must make the same predictions for identical experiments taking place at the same spacetime event in two different inertial frames of reference.On manifolds, the words covariant and contravariant refer to how objects transform under general coordinate transformations. Both covariant and contravariant four-vectors can be Lorentz covariant quantities.
Local Lorentz covariance, which follows from general relativity, refers to Lorentz covariance applying only locally in an infinitesimal region of spacetime at every point. There is a generalization of this concept to cover Poincaré covariance and Poincaré invariance.
I was going through Spacetime Physics by Taylor and Wheeler and came to a point where they showed a proof of Invariance of Spacetime Interval. You can find the proof Here and Here is the second part of that proof.
They used an apparatus that flies straight "up" 3 meters to a mirror. There it...
In this popular science article [1], they say that if our universe resulted to be non-uniform (that is highly anisotropic and inhomogeneous) then the fundamental laws of physics could change from place to place in the entire universe. And according to this paper [2] anisotropy in spacetime could...
Are there non-smooth metrics for spacetime (that don't involve singularities)?
I found this statement in a discussion about the application of local Lorentz symmetry in spacetime metrics:
Lorentz invariance holds locally in GR, but you're right that it no longer applies globally when gravity...
I was reading a discussion where some physicists participated* where the topic of Lorentz invariance violations occurring in cosmology is mentioned.
There, they mention that we can imagine a Lorentz-violating solution to the cosmological equations. What do they mean by that? Can anyone specify...
I have heard that some types of inhomogeneties and topological defects (like cosmic strings) in cosmology have been proposed to be able to break fundamental symmetries of nature such as the Poincaré, Lorentz, diffeomorphism CPT, spatial/time translational...etc symmetries... However, I have not...
I was discussing this paper with a couple of physicists colleagues of mine (https://arxiv.org/abs/2011.12970)
In the paper, the authors describe "spacetimes without symmetries". When I mentioned that, one of my friends said that no spacetime predicted or included in the theory of relativity...
We derive the most basic laws of physics from several fundamental symmetries (those from Noether's theorems, gauge symmetries, Lorentz symmetry...). But are there any types of spacetime where no symmetries, no matter how fundamental, would be valid? Any special metric, geometry or shape?
Before introducing Special Relativity, a textbook highlights the inconsistency of Maxwell's Electrodynamics and Newtonian Mechanics through the standard discussion about the velocity of light in different frames of reference.
A further inconsistency discussed.
In some inertial frame of...
I am reading pretty much everywhere that LET (Lorentz Ether Theory, or call it Neo-Lorentzian Relativity, or whatever theory that involves a preferred undetectable frame with some yet unknown properties that make all the moving objects with respect to this frame length contact and time dilate)...
If ##\partial_{\alpha} J^{\alpha}(x) = 0## then ##Q \equiv \displaystyle{\int} d^3 x J^t(x)## is time-invariant. To show that if ##J^{\alpha}(x)## is a four-vector then ##Q## is also Lorentz-invariant, he re-writes it as \begin{align*}
Q = \int d^4 x J^{\alpha}(x) \partial_{\alpha} H(n_{\beta}...
Hi,
I've read a number of posts here on PF about Einstein's clock synchronization convention.
In the context of SR we know the transformation law between inertial frame's coordinates is actually the Lorentz one. The invariant speed for Lorentz transformation is c (actually it coincides with...
The Lorentz covariance of Maxwell equations was known before Einstein formulated special relativity. So what exactly special relativity brought new with respect to mere Lorentz covariance? Is special relativity just an interpretation of Lorentz invariance, in a sense in which Copenhagen...
Why is that when there is lorentz invariance. Large 3-momentum corresponds to a large energy. And if there was no lorentz invariance. Large 3-momentum does not necessarily need to correspond to a large energy?
What has Lorentz invariance got to do with 3-momentum having large energy or not?
We're trying to reduce the tensor integral ##\int {\frac{{{d^4}k}}{{{{\left( {2\pi } \right)}^4}}}} \frac{{{k^\mu }{k^\nu }}}{{{{\left( {{k^2} - {\Delta ^2}} \right)}^n}}}{\rm{ }}## to a scalar integral (where ##{{\Delta ^2}}## is a scalar). We're told that the tensor integral is proportional...
Homework Statement
I have an assignment to prove that specific intensity over frequency cubed is Lorentz invariant. One of the main tasks there is to prove the invariance of phase space d^3q \ d^3p and I am trying to prove it with symplectic geometry. I am following Jorge V. Jose and Eugene J...
I have an assignment to show that specific intensity over frequency cubed \frac{I}{\nu^3}, is Lorentz invariant and one of the main topics there is to show that the phase space is Lorentz invariant. I did it by following J. Goodman paper, but my professor wants me to show this in another way...
I have a really naive question that I didn't manage to explain to myself. If I consider SUSY theory without R-parity conservation there exist an operator that mediates proton decay. This operator is
$$u^c d^c \tilde d^c $$
where ##\tilde d## is the scalar superpartner of down quark. Now...
Hi there,
I just saw some lectures where they claim that the Klein Gordon equation is the lowest order equation which is Lorentz invariant for a scalar field.
But I could easily come up with a Lorentz invariant equation that is first order, e.g.
$$
(M^\mu\partial_\mu + m^2)\phi=0
$$
where M is...
As I understand it Horava Lifschitz theory breaks lorentz invariance at high energies.
Does this mean we should see photons from gamma ray bursts leave a signal of varying speeds of light for different frequencies?
I am working through Lessons in Particle Physics by Luis Anchordoqui and Francis Halzen; the link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf. I am on page 11, equation 1.3.20. The authors have defined an operator ##L_{\mu\nu} = i( x_\mu \partial \nu - x_\nu \partial \mu)##...
I've commonly heard it said that Lorentz invariance is equivalent to saying that special relativity is obeyed, although I also recall discussions arguing that this is not precisely and technically correct, although the two concepts heavily overlap.
I also understand that Lorentz invariance has...
Hi all,
Clarification question: I've read that string theory is manifestly Lorentz invariant - however, I'm confused about this being true in 4D spacetime or in the full 10D setting of the theory (well, one version anyway). At some point I'd also read in a paper that 4D Lorentz invariance...
Hello! I started reading stuff on QFT and it seems that one of the main points is for the Lagrangian to be Lorentz invariant, so that the equations of motion remain the same in all inertial reference frames. I am not sure however i understand how do non inertial reference frames come into play...
Homework Statement
Show that ##d^4k## is Lorentz Invariant
Homework Equations
[/B]
Under a lorentz transformation the vector ##k^u## transforms as ##k'^u=\Lambda^u_v k^v##
where ##\Lambda^u_v## satisfies ##\eta_{uv}\Lambda^{u}_{p}\Lambda^v_{o}=\eta_{po}## , ##\eta_{uv}## (2) the Minkowski...
Homework Statement
[/B]
The problem is as follows: in a reference frame there is one electron at rest and one incoming positron which annihilates with the electron. The positron energy is E and two gamma rays are produced. Find first the energy of the photons in the center of mass frame as...
Why is it that introducing a hard cut-off ##p^{2}=\Lambda^{2}## breaks Lorentz invariance? Is it simply that it introduces an energy scale and energy is not a Lorentz invariant quantity?
Sorry if this is a trivial question, but I just want to make sure I understand the reasoning as I've...
is spacetime Lorentz invariant like the quantum vacuum?
They say the quantum vacuum is Lorentz invariant.. you can't locate it at any place.. but if spacetime manifold is also Lorentz invariant and you can't locate it at any place.. how come the Earth can curve the spacetime around the Earth...
Consider the following Lagrangian:
##YHLN_{1}^{c} + Y^{c}H^{\dagger}L^{c}N_{1} + \text {h.c.},##
where ##L=(N_{0}, E')## and ##L^{c} = (E^{'c}, N_{0}^{c})## are a pair of ##SU (2)## doublets and ##N_{1}## and ##N_{1}^{c}## are a pair of neutral Majorana fermions...
Hi all,
Some recent comments from the forums here led me to do a bit of reading on the holographic principle, and to a posting on "The Reference Frame" by Lubos Moti about the (likely lack of) 'holographic noise' in the experiment by Craig Hogan at Fermilab...
One more question before Santa comes. There are a number of different related threads, so hopefully I'm not repeating this - however, I haven't found a crisp answer yet.
If one introduces a UV cutoff in the vacuum energy (in an attempt to avoid having infinite vacuum energy), is it possible at...
I read Lucien Hardy's paper whose tittle was "Quantum Mechanics, Local Realistic Theories, and Lorentz Invariant Relativistic Theories". There, he argued that lorentz invariant observables which involved locality assumption contradict quantum mechanics.
I tried to follow his argument, but got...
Homework Statement
For a plane, monochromatic wave, define the width of a wavefront to be the distance between two points on a given wavefront at a given instant in time in some reference frame. Show that this width is the same in all frames using 4-vectors and
in-variants.
Homework...
Homework Statement
I'm asked to prove that Et - p⋅r = E't' - p'⋅r'
Homework Equations
t = γ (t' + ux')
x = γ (x' + ut')
y = y'
z = z'
E = γ (E' + up'x)
px = γ (p'x + uE')
py = p'y
pz = p'z
The Attempt at a Solution
Im still trying to figure out 4 vectors. I get close to the solution but I...
As I understand it, since space-time is modeled as a four dimensional manifold it is natural to consider 4 vectors to describe physical quantities that have a direction associated with them, since we require that physics should be independent of inertial frame and so we should describe it in...
I am looking for a proof that the Feynman propagator is locally a lorentz invariant (at least for scalar fields) also in curved space-times if the background geometry is smooth enough.
I mean, since it is of course a lorentz invariant on flat spaces, this should also be true if a choose a...
Homework Statement
I am meant to show that the following equation is manifestly Lorentz invariant:
$$\frac{dp^{\mu}}{d\tau}=\frac{q}{mc}F^{\mu\nu}p_{\nu}$$
Homework Equations
I am given that ##F^{\mu\nu}## is a tensor of rank two.
The Attempt at a Solution
I was thinking about doing a Lorents...
I recently had someone ask me why we use 4-vectors in special relativity and what is the motivation for introducing them in the first place. This is the response I gave:
From Einstein's postulates( i.e. 1. the principle of relativity - the laws of physics are identical (invariant) in all...
I would like to prove the Lorentz invariance of the Klein-Gordon equation by proving the invariance of the action ##\mathcal{S} = \int d^{4}x\ \mathcal{L}_{KG}## under a Lorentz tranformation.
I would like to do this by first proving the Lorentz invariance of the ##\mathcal{L}_{KG}## and then...
Homework Statement
1. Show directly that if ##\varphi(x)## satisfies the Klein-Gordon equation, then ##\varphi(\Lambda^{-1}x)## also satisfies this equation for any Lorentz transformation ##\Lambda##.
2. Show that ##\mathcal{L}_{Maxwell}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}## is invariant under...
Consider the Heaviside function ##\Theta(k^{0})##.
This function is Lorentz invariant if ##\text{sign}\ (k^{0})## is invariant under a Lorentz transformation.
I have been told that only orthochronous Lorentz transformations preserve ##\text{sign}\ (k^{0})## under the condition that ##k## is a...
I understand that in order to preserve the inner product of two four vectors under a change of coordinates x^{\mu}\rightarrow x^{\mu^{'}}=\Lambda^{\mu^{'}}_{\,\, \nu}x^{\nu} the Minkowski metric must transform as \eta_{\mu^{'}\nu^{'}}=\Lambda^{\alpha}_{\,\...
A well known math theorem says that - if the spatial dimension is odd - D'Alembert equation gives rise to a solution containing a term which is completely supported on the light cone.
A mathematical wrap up could be the following:
"in dimension 3 (and in fact, for all odd dimensions), the...
Hello,
I am re-reading a book about quantum physics and general relativity. To introduce representation of the lorentz group, they explain the definition of lorentz group as the group of transformation that let x² + y² ... -t² unchanged.
But in cuved space the distance is not the same as in...
Hi all,
Just doing some hobby physics while I put off working on my research. In one dimension, the function
\begin{equation}
f(a,b)=[1-\exp(-(a-b)^2)]
\end{equation}
vanishes when a=b. In Minkowski spacetime though, such a function is not so easy to find (if you require Lorentz invariance). If...