See discussions, stats, and author profiles for this publication at: https://www.researchgate.

net/publication/315669958

LECTURES ON TORSION: V - Lorentz Tensor of Curvature

Presentation · March 2017

DOI: 10.13140/RG.2.2.17952.17923

CITATIONS READS

0 104

1 author:

Luca Fabbri
Università degli Studi di Genova
160 PUBLICATIONS 1,614 CITATIONS

SEE PROFILE

All content following this page was uploaded by Luca Fabbri on 28 March 2017.

The user has requested enhancement of the downloaded file.

 LECTURES ON TORSION:
V - Lorentz Tensor of Curvature

Luca Fabbri
LPSC, Grenoble, France
(Dated: March 28, 2017)
This is a series of lectures about torsion gravity in presence of spinors, as given at the LPSC
of Grenoble in France, during the spring of 2017: in the fth lecture, we introduce the Lorentz
formalism by translating all we have done so far.

I. FROM COORDINATE TO LORENTZ transformations. In doing so it may seem that we did not
TENSORS gain much, but having coordinate transformations fully
converted into Lorentz transformations is an advantage,
In the rst part, we have given the general denition of since Lorentz transformations have a very specic form.
a coordinate tensor and the way to move its coordinate Such form can be made explicit: any continuous trans-
indices; coordinate indices are important since they are formation is writable according to a perturbative expan-
the type of indices involved in dierentiation, but on the sion in products of the innitesimal parameters times
other hand, tensors in coordinate indices always feel the their generators; with Λ = I + δG we get that δGη must
specicity of the coordinate system: a tensorial equation be antisymmetric, and we know that 4-dimensional anti-
remains formally the same in all coordinate system, but symmetric matrices have 6 degrees of freedom. Therefore
the tensors themselves change in content while changing Λ = e2σ
1 ab
θab
the coordinate system. The only type of tensors which,
also in content, remain the same in all of the coordinate in which θab = −θba and σab = −σba amount precisely to
systems are the scalars; they are disposal in order to build 6 parameters and 6 generators, themselves verifying
a formalism in which tensors can be made, both in form
and in content, completely invariant. And this formalism [σab , σcd ] = ηad σbc − ηac σbd + ηbc σad − ηbd σac
is what is known to be the Lorentz formalism. and given according to
In Lorentz formalism, the idea is that of introducing a
basis of vectors ξaα having two types of indices: one type (σab )ij = δai ηjb −δbi ηja
of indices (Greek) is the usual coordinate index referring as the real representation: this form is the compact way
to the component of the vector, whereas the other type of of writing the explicit expressions obtained by consider-
indices (Latin) is a new Lorentz index referring to which ing that ησab are 6 antisymmetric matrices, and thus
vector of the basis we are considering; now take a generic
vector Tα and multiply it by vector ξaα contracting the
   
0 100 0 010 0 001
 
coordinate indices together: the result Tα ξaα = Ta is an 1 0 0 0  02  0

0 0 0  03  0
 
0 0 0

object that according to a coordinate transformation law
 
σ 01 = 
0 , σ = , σ = 
0 0 0

1 0 0 0 0 0 0 0
does not transform, thus it is completely invariant, which
is exactly what we wanted. And similarly, tensors with 0 000 0 000 1 000
upper indices must be converted into this formalism as are the generators of the boosts while
well by taking into account by introducing a dual basis of
covectors ξαa as expected: converting a coordinate index
     
0 00 0 0 0 00 0 0 0 0
to a Lorentz index and then back to a coordinate index 0

0 0 0  31 0
 
0 0 1 12 0
 
0 −1 0

requires that ξbα ξαc = δbc and ξkα ξσk = δσα as two consistency σ 23=
0
, σ =
0 0 −1 0
, σ =
0 0 0 0 1 0 0

conditions. Finally, the operation for moving Lorentz in-
dices is performed in terms of the metric tensor in Lorentz 0 01 0 0 −1 0 0 0 0 0 0
form gασ ξaα ξsσ = gas but as we can always ortho-normalize are the generators of the rotations, verifying the commu-
the basis, the metric in Lorentz form is the Minkowskian tation relationships given by the explicit expressions
matrix gas = ηas as it is very well known indeed.
Once the basis ξaσ is assigned, we may pass to another [σ01 , σ02 ] = −σ12 [σ31 , σ12 ] = σ23
basis ξa0σ linked to the initial according to the transfor- [σ02 , σ03 ] = −σ23 [σ12 , σ23 ] = σ31
mation ξa0σ = Λba ξbσ but Λba is chosen as to preserve the [σ03 , σ01 ] = −σ31 [σ23 , σ31 ] = σ12
structure of the Minkowskian matrix and so such that it
has to give η = ΛηΛT known as Lorentz transformation among boosts and among rotations with
and justifying the name of this formalism: after that the [σ31 , σ03 ] = σ01 [σ31 , σ01 ] = −σ03
coordinate tensors are converted into the Lorentz tensors,
they are scalars under coordinate transformations, as we [σ12 , σ01 ] = σ02 [σ23 , σ03 ] = −σ02
have said above, but they are tensors under the Lorentz [σ23 , σ02 ] = σ03 [σ12 , σ02 ] = −σ01
 among dierent boosts and rotations; as a consequence, the fact that coordinate tensors have Greek indices while
by calling θ01 = ϕ1 , θ02 = ϕ2 , θ03 = ϕ3 for the rapidities Lorentz tensors have Latin indices.
and θ23 = −θ1 , θ31 = −θ2 , θ12 = −θ3 for the angles, it is Also algebraic operations hold analogously.
possible to compute the explicit transformations as Given functions Ωabµ such that under a general coordi-
nate transformation transforms as a co-vector, and under
a Lorentz transformation transforms according to
 
cosh ϕ1 − sinh ϕ1 00
 − sinh ϕ1 cosh ϕ1 0 0
 
ΛB1 =  
0 0 1 0
(2)
 0
a0
Ω0a −1 a
)k (∂ν Λ)kb (Λ−1 )bb0
 a 
0 0 01 b0 ν = Λa Ωbν − (Λ
 
cosh ϕ2 0 − sinh ϕ2 0
0 1 0 0 is called spin connection, and no torsion can be dened
 
ΛB2 = 

 − sinh ϕ2 0 cosh ϕ2 0 

as no transposition of indices of dierent types is dened.
0 0 0 1 In terms of it we have that
 
cosh ϕ3 0 0 − sinh ϕ3
Pk=i
0 10 0 Dµ Tra11...r
...ai
= ∂µ Tra11...r
...ai
+ k=1 Ωapµk Tra11...r
...p...ai
−
 
ΛB3 =  (3)
  j j j
 Pk=j p a1 ...ai
 0 01 0  − k=1 Ωrk µ Tr1 ...p...rj
− sinh ϕ3 0 0 cosh ϕ3

for the boosts and is the covariant derivative of the tensor in Lorentz form.
 
1 0 0 0
0 1 0 0 
 
II. LORENTZ CURVATURE
ΛR1 =  
0 0 cos θ1 sin θ1 
0 0 − sin θ1 cos θ1 In Lorentz formalism, from the spin connection we get
 
1 0 0 0
0 cos θ2 0 − sin θ2 
 
ΛR2 =   Gabαβ = ∂α Ωabβ − ∂β Ωabα + Ωakα Ωkbβ − Ωakβ Ωkbα (4)
0 0 1 0 
0 sin θ2 0 cos θ2
called curvature tensor again; since we already dened a
 
1 0 0 0
0

cos θ3 sin θ3 0 
 curvature tensor, this may very well look like a redundant
ΛR3 =   denition, but in the next lesson we will prove that the
0 − sin θ3 cos θ3 0  two denitions are actually equivalent.
0 0 0 1 From this we have that
for the rotations, such that any product of these specic
Lorentz transformations gives the full form of the Lorentz [Dµ , Dν ]Tra11...r
...ai
= Qηµν Dη Tra11...r
...ai
+
transformation, which we will employ next. Pk=i ajk j

Once the Lorentz transformation is assigned, we can

a1 ...p...ai
+ k=1 G pµν Tr1 ...rj − (5)
re-write the geometrical background exactly as we did
Pk=j
− k=1 Gprk µν Tra11...p...r
...ai
j
before, starting from the fact that given a Lorentz trans-
formation Λ the set of functions Tra11...r
...ai
j
transforming as
is the commutator of covariant derivatives of tensors in
Lorentz formalism as it would be expected.
0a0 ...a0
Tr0 ...r
a0
= (Λ−1 )rr10 ...(Λ−1 )rrnn0 (Λ)a11 ...(Λ)am
a0
a1 ...am (1) Additionally, we also have that
m Tr1 ...rn
1 m
0
1 n 1

is called tensor in Lorentz form ; the denition of scalars Dµ Gajκρ + Dκ Gajρµ + Dρ Gajµκ +
(6)
and vector are given similarly; also the properties of sym- +G jβµ Qβ ρκ + Gajβκ Qβ µρ + Gajβρ Qβ κµ
a
≡0
metry and transposition, as well as contractions and irre-
ducibility, are given in an analogous manner. So far the
only thing that dierentiate the two forms of tensors are as the Jacobi-Bianchi identity in Lorentz form.

View publication stats