Download Flat spin connections in the Teleparallel equivalent of General

Flat spin connections in the Teleparallel equivalent of General
Relativity
A. Zubiaga∗
Abstract
arXiv:1705.06585v1 [gr-qc] 18 May 2017
This work shows that within the Teleparallel equivalent of General Relativity the flat spin connection has a kinematical character, decoupled from the dynamics of the tetrad and the matter
fields. A change of the flat spin connection introduces a global acceleration field in the chosen
frame. The inertial effects introduced this way can always be reversed by an affine transformation
of the flat spin connection which will cancel the inertial effects without changing the dynamics of
the gravitationally coupled fields. The tetrad remains unchanged but the gravitational potential,
the source of anholonomy, is modified by a global acceleration field. This formalism shows gauge
invariance under local translation transformations. In addition, it is invariant under global acceleration fields introduced by local Lorentz affine transformations of the flat spin connection. The
explicit separation of the gravitation-matter coupling from reference frame related inertial effects
can ease the treatment of the constraints of gravitational systems. Further studies are needed to
unveil the character of the internal constraints of the theory and explore its canonical quantization.
∗
Electronic address: [email protected]
1
I.
INTRODUCTION
The description of gravitational interaction was revolutionised after A. Einstein proposed
his General Relativity (GR) theory [1]. The theory has been tested to great detail in
its most important predictions like the the deflection of light [2], the precession of the
perihelion of Mercury [1], the gravitational redshift of light [3], and the recent measurement
of the gravitational waves emitted by two black-holes merger [4]. Still, important challenges
remain related to the quantum formulation of the theory. The loop quantum gravity [5] and
spin foams [6] are promising candidates for a quantum theory of gravitation and the string
theory [7] promises to unify the description of the four fundamental interactions. However,
the research program followed by both approaches have evolved with difficulties and many
open questions remain. Deepening in the knowledge of the structure of GR may lead to
the formulation of a finite quantum theory for gravitation phenomena. Here a Teleparallel
formulation is introduced based in the tetrad form and a flat spin connection. Blagojević
et al. [8] also explored the role of flat spin connections in GR using the canonical analysis.
The new formulation is characterized by an invariance of the gravitational coupling under
local Poincare transformations.
Throughout the paper, latin {a, b, c, ...} and greek {µ, ν, ρ, ...} letters are reserved for indices of the Minkowski and the Riemann tangent bundles, respectively. Section II describes
the main characteristics of the Teleparalell formulation of GR with flat spin connections.
Section III introduces an affine transformation of the flat spin connection that leaves invariant the dynamics of the gravitational system and shows the kinematical character of the flat
spin connection. It is also remarked that the affine transformation represents non-inertial
changes of reference frames. Section IV describes the gauge properties of the gravitational
potential.
II.
TELEPARALELL GRAVITY WITH FLAT SPIN CONNECTIONS
The Teleparallel spacetime is composed by an internal algebraic tensor space with a
Minkowski metric ηab and coordinate {X a } and a Riemann spacetime with metric gµν and
coordinate {xµ }. The tangent bundles of both spaces are soldered by the tetrad form
haµ (x). The inverse of the tetrad form haµ (x) is determined by the conditions haµ (x) hbµ (x)
2
= δba and haµ (x) haν (x) = δµν . The anholomic bases for the tangent bundle of the internal
spacetime are {ha } = {haµ ∂µ }, where the right part is written as a function of the coordinate
vector bases {∂µ } of the Riemann spacetime. The bases for the dual forms are written as
{ha } = {haµ dxµ } using the coordinate bases for the covector fields {dxµ } of the Riemann
spacetime. The existence of these frames is guaranteed by the differentiable character of the
spacetime.
The basis of the anholomic vector space {ha } satisfies the commutation relation
[ha , hb ] = f cab hc
(1)
where f cab are the structure coefficients or coefficients of anholonomy for the frame or tetrad
field ha . A tetrad {h0a } for which f 0cab = 0 is said to be holonomic. The metric of the
Riemann and the internal spacetimes are linked by
gµν = ηab haµ hbν ,
(2)
This relation determines the tetrad up to local Lorentz transformations, h0a = Λab (xµ )hb ,
0
which will keep the Riemann metric unchanged (gµν = ηab
h0aµ h0bν ).
A.
Soldered bundles
Space-time forms φµ and Minkowski forms φa are soldered by the tetrad haµ
φµ = haµ φa ,
(3)
and space-time vectors φµ and Minkowski vectors φa by the inverse tetrad haµ
φµ = haµ φa .
(4)
The Fock-Ivanenko derivative [9] Dµ φa , is a covariant derivative defined in the Teleparallel
spacetime using a flat spin connection Aµ which assumes values in the Lie algebra of the
Lorentz group Aµ = 12 Aabµ Ŝab , where Ŝab are a representation of the Lorentz generators and
Aabµ = −Abaµ .
The Fock-Ivanenko covariant derivative for a vector φa is
Dµ φa = ∂µ φa + Aabµ φb .
3
(5)
The soldering condition haρ Dν φa = ∇ν φρ links it to the spacetime covariant derivative
∇ν φρ = ∂ν φρ + Γρµν φµ
(6)
with the Weitzenböck spacetime connection Γρµν
Γρµν = haρ ∂ν haµ + haρ Aabν hbµ
(7)
fulfilling the metric compatibility condition
∂ν haµ + Aabν hbµ − Γβ µν haβ = 0.
(8)
An affine transformation of the Lorentz connection by a tensor field Gabµ = −Gbaµ defines
another Lorentz connection Ãabµ = Aabµ + Gabµ and spacetime connection Γ̃αβµ = Γαβµ +
hbβ haα Gabµ which is also metric compatible with the tetrad field.
For a given local Lorentz transformation in the Minkowski tangent vector space
φ0a = Λab (x)φb
(9)
the tetrad field and the Fock-Ivanenko derivative transform covariantly.
h0aµ = Λab (x)hbµ
(10)
Dµ φ0a = Λab (x)Dµ φb
(11)
and the connections acquires a vacuum component
A0abµ = Λac (x)Λbd (x)Acdµ + Λac (x)∂µ Λbc (x)
(12)
The transformation does not affect the spacetime vectors as shown below for the spacetime
metric.
0
gµν
= ηcd Λac (x)Λbd (x)Λae (x)Λbf (x)heµ hf ν = ηcd hcµ hdν = gµν
(13)
The Fock-Ivanenko derivative can be defined for a field ψ with arbitrary spin as
Dµ ψ = ∂µ ψ + Aabµ Ŝab ψ.
4
(14)
B.
Curvature and Torsion
The Lorentz connection Aabµ has zero curvature,
Rabµν = ∂µ Aabν − ∂ν Aabµ + Aacµ Acbν − Aacν Acbµ = 0,
(15)
but non zero torsion
T aµν = Dµ haν − Dν haµ
(16)
The tetrad can be written in terms of a trivial tetrad, Dν eaµ = 0 with null torsion, and
an anholomic term B aµ
haµ = eaµ + B aµ
(17)
In non-trivial tetrads, Bµa accounts for the anholonomy of the tetrad field haµ and it is the
source of torsion.
T aµν = Dµ B aν (x) − Dν B aµ (x)
C.
(18)
Equation of motion of free falling particles
The equation of motion of a test particle moving with speed ua in the Teleparallel spacetime can be derived from the principle of least action [10]:
dua
+ Aabµ − K abµ ub uµ = 0.
ds
(19)
where s is an affine parameter proportional to the proper time of the test particle and
the contortion K abµ = 1/2 Tb aµ + Tµ ab − T abµ plays the role of an external force. The
equivalence of equation (19) with the geodesic equation of GR can be shown by noting that
Aabµ − K abµ is the spin connection of GR.
The Fock-Ivanenko covariant derivative describes the kinematic properties of a free point
particle moving in the Teleparallel spacetime for a given reference frame (Σx , Aabµ ) in the
absence of anholonomy terms
dua
+ Aabµ ub uµ = 0
ds
Dµ u a = 0
5
(20)
(21)
For an inertial reference frame Aabµ = 0 without an anholomic term, B aµ (x) = 0 the equation
of motion of a point particle reduces to
dua
= 0.
ds
(22)
which describes a particle with constant proper speed moving in one dimension, as expected
in the absence of forces.
D.
Lagrangian formulation
The Lagrangian density of the Teleparalell equivalent of GR
c4 h
1 ρ
1 ρ νµ
µν
ρ
νµ
LG =
T T
+ T µν T ρ − T µρ T ν
16πG 4 µν ρ
2
(23)
can be shown to be equal to the Lagrangian density of GR up to a divergence surface
terms [10]. The Lagrangian density is includes the tetrad haµ , Aabµ and ∂ν haµ . There are
not kinetic terms for the flat spin connection ∂ν Aabµ . The counterpart of the equation of
Einstein is:
∂σ (hSa ρσ ) − khρa = kΘaρ
(24)
Sa ρσ is the superpotential
with K ρσa
Sa ρσ = K ρσa − hσa T θρθ + hρa T θσθ
(25)
= 1/2 T σρa + Ta ρσ − T ρσa the contortion. The gauge current aρ is defined as
ρa =
hρ
1
1 λ νρ c
ha Sc T νλ − a LG + Acaσ Sc ρσ
k
h
k
(26)
with k = 8πG/c4 and hΘaρ = −δLM /δhaρ the matter energy-momentum tensor. The gauge
current ρa has a pseudotensor term k1 Acaσ Sc ρσ which describes the non-inertial effects and
can be eliminated by choosing an inertial reference frame. A conservation law exists for Θaρ
and aρ
∂ρ (khρa + kΘaρ ) = 0,
(27)
that can be interpreted as a conservation law for the matter and the gravitational potential
together.
A Lagrange multiplier that explicitly forces the flatness of the spin connection (LR =
(c4 h/16πG) λabµν Rabµν ) adds a kinetic term for Aabµ . Its Euler-Lagrange equations determine
6
the Lagrange multipliers λabµν
Dν λabνµ − λabµν + Γρνρ − K ρνρ λabνµ − λabµν = 2 K µba + δab K ρµρ − haµ K ρbρ
(28)
where Γρνρ is the Weitzenböck spacetime connection. This equation remains valid when
matter fields under the minimal coupling prescription are included.
E.
Gravitational coupling of matter fields
Gravitational coupling of a matter field φi must be compatible with the equivalence
principle. The minimal coupling prescription requires that the derivative coupling is done
through the spin connection of GR Ǎabµ = Aabµ − K abµ . The resulting covariant derivative
is
ϵ φi = ∂µ φi + Aabµ − K abµ Ŝab φi
(29)
with the contortion K abµ = 1/2 Tb aµ + Tµ ab − T abµ and the spin operator Ŝab acting on
the field φi of spin s. The Lagrangian of matter fields LM φi , ∂ν φi , Aabµ , haµ , ∂ν haµ can
thus be written as a function of the fields and its derivatives (φi , ∂ν φi ), the tetrad and
its derivatives (haµ , ∂ν haµ ). However, only the Lorentz connection Aabµ appears but the
functional derivative
δLM
δLM δ Acdµ − K cdµ
Ŝcd φi = 0
=
a
i
ˇ
δAabµ
δA
δDν φ
bµ
(30)
is zero thanks to the anti-symmetry of Ŝcd . This result guarantees that the Aabµ remains
undertermined by the field equations.
III.
INVARIANCE UNDER AFFINE TRANSFORMATIONS
The effect of affine transformations within the subspace of the flat spin connections
Aabµ → Áabµ = Aabµ + Gabµ generated by a tensor Gabµ will help to understand the role
of Aabµ . The antisymmetry properties of the spin connections forces that the transformation
tensor is antisymmetric in the internal indexes Gabµ = −Gbaµ . The new flat spin connection
fulfills the metric compatibility condition of equation (8).
After the affine transformation the Fock-Ivanenko derivative of the new reference frame
Σx , Áabµ changes following equations (17) and (18). The new torsion is
T́ aµν = T aµν + Gabµ hbν − Gabν hbµ
7
(31)
or
T́ aµν = T aµν + Aabµ αbν + Gabµ B bν + αbν
−Aabν αbµ − Gabν B bµ + αbµ
(32)
where αµa is a global acceleration field. Given Gabµ and using the Fock-Ivanenko covariant
derivative of the trivial tetrad is zero for the original frame Dν eaµ = 0 and for the transformed
frame D0ν éaµ = 0 an equation determining αµa can be written. Writing the transformed trivial
tetrad as e0aµ = eaµ − αaµ the equation reads
Dν αaµ = Gabν ebµ − αbµ ,
(33)
which does not depend on the tetrad or the gravitational potential B aµ . The new contortion
is remarkably simple thanks to the antisymmetry of Gabµ
Ḱ abµ = K abµ + Gabµ
(34)
The spin coefficient of GR Ǎabµ = Aabµ − K abµ remains invariant under the affine transformation what implies that the coupling to matter is neither modified. The invariance of the
field equations can be shown to be invariant by rewriting them as the Einsteins equation
1 ρ
ρ
h Řa − ha Ř = kΘaρ
(35)
2
where Řabµν is Riemann tensor of the spin connection of GR Ǎabµ .
A.
Example: Uniformly accelerated observer
An uniformly accelerated particle along cartesian direction x1 in the flat spacetime is
described by an inertial observer as
xµ = a−1 sinh(aτ ), a−1 cosh(aτ ), 0, 0
(36)
where x0 is the time coordinate. The proper speed and acceleration are
uµ = (cosh(aτ ), sinh(aτ ), 0, 0)
(37)
du
= (a sinh(aτ ), a cosh(aτ ), 0, 0) ≡ Γ̃µν uν
dτ
(38)
µ
8
In the tangent space the equation for dua /dτ is
dua
= Ξab ub
dτ
(39)
where a trivial tetrad field eaµ = δµa has been used. For a particle with trajectory xa (τ ) the
partial derivatives of Eq. (39) can be written
∂µ ua = Ξabµ ub
(40)
where Ξabµ = ∂µ Ξab . Since the reference frame is inertial Āabµ = 0, the partial derivative is
also a Fock-Ivanenko covariant derivative
Dµ ua = Ξabµ ub .
(41)
and Ξabµ is the contortion of a gravitational potential B aµ . Using that Ξabµ represents a
global acceleration field and the affine character of the space of Lorentz connections, the
equation for a uniformly accelerated particle can be written as
D0µ ua = ∂µ ua + A0abµ ub = 0.
(42)
where A0abµ = Āabµ − Ξabµ is also a Lorentz connection. The reference frame is not inertial
anymore and the particle is now described as moving in the absence of forces. There exists
a third reference frame where the particle is moving in the absence of an external force with
uniform velocity. This reference frame is defined by changing the coordinate system so that
A0abµ = 0. In the presence of matter fields, the external forces cannot be made zero, but the
inertial reference frame would be valid to describe the motion of the particle.
B.
Equivalence of accelerated observers
In a Minkowski spacetime inertial observers have a privileged status because only the
Lorentz transformations between inertial frames, and not transformations between inertial
and non-inertial reference frames are isometries. However, the separation between inertial
and non-inertial observers is problematic. Inertial observers are typically defined in a rather
abstract way as being inertial respect to the center of mass of the whole universe, or, in a
more practical way, as being in rest respect to the distant stars. The equivalence principle of
Einstein allows treating all observers as equivalents. A. Einstein introduced the equivalence
9
between inertial and gravitational mass as the cornerstone of his geometrical theory of
gravity. He illustrated it using a gedankenexperiment which showed that the law of physics
inside a closed room will be the same when it moves with constant acceleration and when
it is placed inside a constant gravitational field [1]. An observer inside the closed room will
not be able to say whether he is static inside a gravitational field or moving with constant
acceleration. The gedankenexperiment can be modified to include arbitrarily accelerated
observers. The accelerated observer is equivalent to an inertial observer in a non-constant
gravitation field. The gravitation field defined this way will be called an acceleration field,
to account to the property that its Riemann curvature is zero. Respect to this observer,
objects in free fall will move in an accelerated trajectory. The equivalence principle ensures
that there are observers, non-inertial with respect to the original one, which measure a
zero gravitational field globally. In that reference frame, objects in free fall will move with
constant speed or they will remain at rest.
To illustrate the contradiction of giving to inertial observers an absolute character and,
at the same time keeping the equivalence principle, two observers moving with an arbitrary
mutual acceleration in an empty (flat) spacetime will be considered. When observer A is
inertial, it can determine that the measurements of its rulers and clocks are determined by
the geometry of a Minkowski spacetime and it will measure that observer B is accelerated.
Using the transformation laws of Special Relativity, it will find that the measurements of
observer B is not inertial and by doing a local Lorentz coordinate transformation will define
a reference frame where observer B is a non-inertial rest observer. There is not a coordinate
transformation where observer B will be inertial. However, the symmetry between both
observers allow to consider observer B to be inertial. Then, the same arguments of the
previous case can be used to find that observer A will be accelerated and it cannot be made
inertial by any coordinate transformation.
This paradox can be solved within the Teleparallel formulation of GR introducing a
flat spin connection. The observers are described by reference frames described by a flat
connection Aabµ and a coordinate system Σx . Given Aabµ , only reference frames with a
set of coordinate systems Σx0 and its global Lorentz coordinate transformations are iner
tial and the rest include non-inertial effects. Two reference frames Σx , Aabµ + GAabµ and
Σx , Aabµ + GB abµ can be defined for observers A and B which differ by a global acceleration
field αB aµ −αAaµ . Both reference frames cannot be made inertial by the same coordinate sys10
tem. However, the affine transformation of the Lorentz connection connects both reference
frames, making them equivalent descriptions of the same physical system.
IV.
GAUGE PROPERTIES OF THE GRAVITATIONAL POTENTIAL
While the gravitational potential B aµ is the dynamical component of the tetrad, the flat
spin connection Aabµ is a pure kinematic field. The antisymmetry of the Lorentz indexes
leaves 24 non-zero components but the null curvature condition Rabµν = 0 further decreases
the number of independent components. It sets 36 constraint equations from which 24 are
seen to be redundant when taking into account the second Bianchi identity Rab[µν,σ] = 0.
The resulting 12 independent constraints leave 12 free components for Aabµ . The invariance
of the Lagrangian under local Lorentz transformations of the internal space indexes requires
6 free components. The other 6 components correspond to the affine transformation Aabµ →
Aabµ + Gabµ where the tensor Gabµ = −Gbaµ = Dµ Λab must be a representation of the Lorentz
group to keep the null curvature condition.
The global acceleration fields introduced in B aµ by the Lorentz affine transformation
come from inertial effects that changes the torsion of the tetrad. However, there are another
transformations which do not change the tetrad nor the torsion given by the gradient Dµ a .
The Lorentz 4-vector a can be shown to be infinitesimal gauge translations [10].
In GR, geometry of the Riemann spacetime is described by the metric and the only invariance comes from diffeomorphism of the coordinate systems. However, the present formalism,
in addition to the invariance under diffeomorphisms of the coordinate systems, it also shows
Poincare invariance, thanks to the kinematical character of the flat spin connection. At
classical level, this difference does not have any major implication and the predictions of
the equation of Einstein and equation 24 are equivalent. However, the natural separation
of the gravitation-matter interaction dynamics from the inertial effects can be beneficial to
understand better the constraints of gravitational systems. The divergences in the matrix
elements coming from the non-bound character of the inertial effects can be treated consistently. This can lead to a renormalizable formulation of the quantum gravity under a frame
formalism.
11
V.
CONCLUSIONS
Flat spin connections have been used within the Teleparallel formulation of GR to treat
consistently the kinematical effects coming from non-inertial frames. It has been shown that
the Euler-Lagrangian equations do not determine the evolution of the flat spin connection.
A transformation of the flat spin connection changes the torsion, but the tetrad and the spin
connection of GR remain invariant. This property allows converting non-inertial observers
into inertial observers without affecting the description of the physical system. The reference
frame for an observer is better defined using a coordinate system and the flat spin connection
where the flat spin connection accounts for the the kinematical state of the observer. A nonzero flat spin connection introduces a global acceleration field and inertial observers are
those for which the flat spin connection is zero. The source anholonomy is the gravitational
potential and it manifests as a gauge field under local Poincare transformations. Local
translation transformations add a torsionless term to the gravitational potential. On the
other hand, affine transformations of the flat spin connection induce an anholonomic gauge
transformation. The natural separation of the gravitation-matter interaction dynamics from
the inertial effects provided by this formalism can provide a natural framework to study
the constraints of gravitational systems. This property can lead to a valid renormalizable
quantum frame gravity theory.
[1] A. Einstein, “Die Grundlage der allgemeinen Relativitätstheorie,” Annalen der Physik,
vol. 354, pp. 769–822, Jan. 1916.
[2] F. W. Dyson, A. S. Eddington, and C. Davidson, “A Determination of the Deflection of Light
by the Sun’s Gravitational Field, from Observations Made at the Total Eclipse of May 29,
1919,” Philosophical Transactions of the Royal Society of London A: Mathematical, Physical
and Engineering Sciences, vol. 220, pp. 291–333, Jan. 1920.
[3] R. V. Pound, “Gravitational Red-Shift in Nuclear Resonance,” Physical Review Letters, vol. 3,
no. 9, pp. 439–441, 1959.
[4] LIGO Scientific Collaboration and Virgo Collaboration, “Observation of Gravitational Waves
from a Binary Black Hole Merger,” Physical Review Letters, vol. 116, p. 061102, Feb. 2016.
[5] A. Ashtekar and J. Lewandowski, “Background independent quantum gravity: a status re-
12
port,” Classical and Quantum Gravity, vol. 21, no. 15, p. R53, 2004.
[6] A. Perez, “The spin-foam approach to quantum gravity,” Living Reviews in Relativity, vol. 16,
p. 3, Dec. 2013.
[7] M. B. Green, J. H. Schwarz, and E. Witten, Superstring theory volume 1. Introduction. Cambridge Monographs on Mathematical Physics, Cambridge University Press, July 2012.
[8] M. Blagojević and M. Vasilić, “Conservation laws in the teleparallel theory of gravity,” Phys.
Rev. D, vol. 64, p. 044010, Jul 2001.
[9] V. Fock and D. Iwanenko, “Über eine mögliche geometrische Deutung der relativistischen
Quantentheorie,” Zeitschrift für Physik, vol. 54, no. 11-12, pp. 798–802, 1929.
[10] R. Aldrovandi and J. G. Pereira, Teleparallel Gravity. Dordrecht: Springer Netherlands, 2013.
13