In relativistic physics, Lorentz symmetry, named for 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".[1]
Lorentz covariance, a related concept, is a property of the underlying
spacetime manifold.
A physical quantity is said to be
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
Contents 1 Examples 1.1 Scalars 1.2 Four-vectors 1.3 Four-tensors 2 Lorentz violating models 3 See also 4 Notes 5 References Examples[edit]
In general, the (transformational) nature of a Lorentz
tensor[clarification needed] can be identified by its tensor order,
which is the number of free indices it has. No indices implies it is a
scalar, one implies that it is a vector, etc. Some tensors with a
physical interpretation are listed below.
The sign convention of the
Δ s 2 = Δ x a Δ x b η a b = c 2 Δ t 2 − Δ x 2 − Δ y 2 − Δ z 2 displaystyle Delta s^ 2 =Delta x^ a Delta x^ b eta _ ab =c^ 2 Delta t^ 2 -Delta x^ 2 -Delta y^ 2 -Delta z^ 2
Δ τ = Δ s 2 c 2 , Δ s 2 > 0 displaystyle Delta tau = sqrt frac Delta s^ 2 c^ 2 ,,Delta s^ 2 >0
L = − Δ s 2 , Δ s 2 < 0 displaystyle L= sqrt -Delta s^ 2 ,,Delta s^ 2 <0 Rest mass m 0 2 c 2 = P a P b η a b = E 2 c 2 − p x 2 − p y 2 − p z 2 displaystyle m_ 0 ^ 2 c^ 2 =P^ a P^ b eta _ ab = frac E^ 2 c^ 2 -p_ x ^ 2 -p_ y ^ 2 -p_ z ^ 2 Electromagnetism invariants F a b F a b = 2 ( B 2 − E 2 c 2 ) G c d F c d = 1 2 ϵ a b c d F a b F c d = − 4 c ( B → ⋅ E → ) displaystyle begin aligned F_ ab F^ ab &= 2left(B^ 2 - frac E^ 2 c^ 2 right)\G_ cd F^ cd &= frac 1 2 epsilon _ abcd F^ ab F^ cd =- frac 4 c left( vec B cdot vec E right)end aligned D'Alembertian/wave operator ◻ = η μ ν ∂ μ ∂ ν = 1 c 2 ∂ 2 ∂ t 2 − ∂ 2 ∂ x 2 − ∂ 2 ∂ y 2 − ∂ 2 ∂ z 2 displaystyle Box =eta ^ mu nu partial _ mu partial _ nu = frac 1 c^ 2 frac partial ^ 2 partial t^ 2 - frac partial ^ 2 partial x^ 2 - frac partial ^ 2 partial y^ 2 - frac partial ^ 2 partial z^ 2 Four-vectors[edit] 4-displacement Δ X a = ( c Δ t , Δ x → ) = ( c Δ t , Δ x , Δ y , Δ z ) displaystyle Delta X^ a =left(cDelta t, vec Delta x right)=(cDelta t,Delta x,Delta y,Delta z) 4-position X a = ( c t , x → ) = ( c t , x , y , z ) displaystyle X^ a =left(ct, vec x right)=(ct,x,y,z) 4-gradient which is the 4D partial derivative: ∂ a = ( ∂ t c , − ∇ → ) = ( 1 c ∂ ∂ t , − ∂ ∂ x , − ∂ ∂ y , − ∂ ∂ z ) displaystyle partial ^ a =left( frac partial _ t c ,- vec nabla right)=left( frac 1 c frac partial partial t ,- frac partial partial x ,- frac partial partial y ,- frac partial partial z right) 4-velocity U a = γ ( c , u → ) = γ ( c , d x d t , d y d t , d z d t ) displaystyle U^ a =gamma left(c, vec u right)=gamma left(c, frac dx dt , frac dy dt , frac dz dt right) where U a = d X a d τ displaystyle U^ a = frac dX^ a dtau 4-momentum P a = ( m c , p → ) = ( E c , p → ) = ( E c , p x , p y , p z ) displaystyle P^ a =left(mc, vec p right)=left( frac E c , vec p right)=left( frac E c ,p_ x ,p_ y ,p_ z right) where P a = m o U a displaystyle P^ a =m_ o U^ a 4-current J a = ( c ρ , j → ) = ( c ρ , j x , j y , j z ) displaystyle J^ a =left(crho , vec j right)=left(crho ,j_ x ,j_ y ,j_ z right) where J a = ρ o U a displaystyle J^ a =rho _ o U^ a 4-potential A a = ( ϕ / c , A → ) = ( ϕ / c , A x , A y , A z ) displaystyle A^ a =left(phi /c, vec A right)=left(phi /c,A_ x ,A_ y ,A_ z right) Four-tensors[edit] Kronecker delta δ b a = 1 if a = b , 0 if a ≠ b . displaystyle delta _ b ^ a = begin cases 1& mbox if a=b,\0& mbox if aneq b.end cases
η a b = η a b = 1 if a = b = 0 , − 1 if a = b = 1 , 2 , 3 , 0 if a ≠ b . displaystyle eta _ ab =eta ^ ab = begin cases 1& mbox if a=b=0,\-1& mbox if a=b=1,2,3,\0& mbox if aneq b.end cases Levi-Civita symbol ϵ a b c d = − ϵ a b c d = + 1 if a b c d is an even permutation of 0123 , − 1 if a b c d is an odd permutation of 0123 , 0 otherwise. displaystyle epsilon _ abcd =-epsilon ^ abcd = begin cases +1& mbox if abcd mbox is an even permutation of 0123 ,\-1& mbox if abcd mbox is an odd permutation of 0123 ,\0& mbox otherwise. end cases
F a b = [ 0 1 c E x 1 c E y 1 c E z − 1 c E x 0 − B z B y − 1 c E y B z 0 − B x − 1 c E z − B y B x 0 ] displaystyle F_ ab = begin bmatrix 0& frac 1 c E_ x & frac 1 c E_ y & frac 1 c E_ z \- frac 1 c E_ x &0&-B_ z &B_ y \- frac 1 c E_ y &B_ z &0&-B_ x \- frac 1 c E_ z &-B_ y &B_ x &0end bmatrix Dual electromagnetic field tensor G c d = 1 2 ϵ a b c d F a b = [ 0 B x B y B z − B x 0 1 c E z − 1 c E y − B y − 1 c E z 0 1 c E x − B z 1 c E y − 1 c E x 0 ] displaystyle G_ cd = frac 1 2 epsilon _ abcd F^ ab = begin bmatrix 0&B_ x &B_ y &B_ z \-B_ x &0& frac 1 c E_ z &- frac 1 c E_ y \-B_ y &- frac 1 c E_ z &0& frac 1 c E_ x \-B_ z & frac 1 c E_ y &- frac 1 c E_ x &0end bmatrix Lorentz violating models[edit] See also: Modern searches for Lorentz violation In standard field theory, there are very strict and severe constraints on marginal and relevant Lorentz violating operators within both QED and the Standard Model. Irrelevant Lorentz violating operators may be suppressed by a high cutoff scale, but they typically induce marginal and relevant Lorentz violating operators via radiative corrections. So, we also have very strict and severe constraints on irrelevant Lorentz violating operators. Since some approaches to quantum gravity lead to violations of Lorentz invariance,[2] these studies are part of Phenomenological Quantum Gravity. Lorentz violations are allowed in string theory, supersymmetry and Horava-Lifshitz gravity.[3] Lorentz violating models typically fall into four classes:[citation needed] The laws of physics are exactly
Models belonging to the first two classes can be consistent with
experiment if Lorentz breaking happens at Planck scale or beyond it,
or even before it in suitable preonic models,[6] and if Lorentz
symmetry violation is governed by a suitable energy-dependent
parameter. One then has a class of models which deviate from Poincaré
symmetry near the Planck scale but still flows towards an exact
4-vector Antimatter tests of Lorentz violation Fock–Lorentz symmetry General covariance Lorentz invariance in loop quantum gravity Lorentz-violating electrodynamics Lorentz-violating neutrino oscillations Symmetry in physics Notes[edit] ^ "Framing Lorentz symmetry". CERN Courier. 2004-11-24. Retrieved
2013-05-26.
^ Mattingly, David (2005). "Modern Tests of Lorentz Invariance".
Living Reviews in Relativity. 8 (1): 5. arXiv:gr-qc/0502097 .
Bibcode:2005LRR.....8....5M. doi:10.12942/lrr-2005-5.
PMC 5253993 . PMID 28163649.
^ Neutrino Interferometry for High-Precision Tests of Lorentz Symmetry
with IceCube
^ Luis Gonzalez-Mestres (1995-05-25). "Properties of a possible class
of particles able to travel faster than light". Dark Matter in
Cosmology: 645. arXiv:astro-ph/9505117 .
Bibcode:1995dmcc.conf..645G.
^ Luis Gonzalez-Mestres (1997-05-26). "Absence of
Greisen-Zatsepin-Kuzmin Cutoff and Stability of Unstable Particles at
Very High Energy, as a Consequence of Lorentz Symmetry Violation".
Proceedings of the 25th International Cosmic Ray Conference (held 30
July - 6 August. 6: 113. arXiv:physics/9705031 .
Bibcode:1997ICRC....6..113G.
^ Luis Gonzalez-Mestres (2014). "Ultra-high energy physics and
standard basic principles. Do Planck units really make sense?" (PDF).
EPJ Web of Conferences. EPJ Web of Conferences (ICNFP 2013
Conference). 71: 00062. Bibcode:2014EPJWC..7100062G.
doi:10.1051/epjconf/20147100062.
^ Kostelecky, V.A.; Russell, N. (2010). "Data Tables for Lorentz and
CPT Violation". arXiv:0801.0287v3 .
^ Sanchez, Daniel S., et al. "Discovery of Lorentz-violating type-II
Weyl fermions in LaAlGe." Bulletin of the American Physical Society 62
(2017).
^ Yan, Mingzhe, et al. "Lorentz-violating type-II Dirac fermions in
transition metal dichalcogenide PtTe2." Nature Communications 8
(2017).
^ Deng, Ke, et al. "Experimental observation of topological Fermi arcs
in type-II
References[edit] Background information on Lorentz and CPT violation: http://www.physics.indiana.edu/~kostelec/faq.html Mattingly, David (2005). "Modern Tests of Lorentz Invariance". Living Reviews in Relativity. 8 (1): 5. arXiv:gr-qc/0502097 . Bibcode:2005LRR.....8....5M. doi:10.12942/lrr-2005-5. PMC 5253993 . PMID 28163649. Amelino-Camelia G, Ellis J, Mavromatos NE, Nanopoulos DV, Sarkar S (June 1998). "Tests of quantum gravity from observations of bold gamma-ray bursts". Nature. 393 (6687): 763–765. arXiv:astro-ph/9712103 . Bibcode:1998Natur.393..763A. doi:10.1038/31647. Retrieved 2007-12-22. Jacobson T, Liberati S, Mattingly D (August 2003). "A strong astrophysical constraint on the violation of special relativity by quantum gravity". Nature. 424 (6952): 1019–1021. arXiv:astro-ph/0212190 . Bibcode:2003Natur.424.1019J. doi:10.1038/nature01882. PMID 12944959. Retrieved 2007-12-22. Carroll S (August 2003). "Quantum gravity: An astrophysical constraint". Nature. 424 (6952): 1007–1008. Bibcode:2003Natur.424.1007C. doi:10.1038/4241007a. PMID 12944951. Retrieved 2007-12-22. Jacobson, T.; Liberati, S.; Mattingly, D. (2003). "Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics". Physical Review D. 67 (12): 124011. arXiv:hep-ph/0209264 . Bibcode:2003PhRvD..67l4011J. doi:10.1103/PhysRevD.67.1 |