Pis’ma v ZhETF, vol. 116, iss. 1, pp. 60 - 61

Riemann-Cartan gravity with dynamical signature

S. Bondarenko, M. A. Zubkov¹⁾

Physics Department, Ariel University, 40700 Ariel, Israel

Submitted 18 March 2022

Resubmitted 9 May 2022

Accepted 19

May 2022

DOI: 10.31857/S1234567822130092, EDN: ixkptw

∫

√

Model of Riemann-Cartan gravity with varying sig-

S_O =

d⁴xe

OE^µcE^νf (D_µO)_ab (D_νO)_de α^abc;def with

√

nature of metric is considered. The basic dynamical vari-

= det e =

e^aµe^bνe^cρe^dσǫ_abcdǫ^µνρσ,

O =

det O .

ables of the formalism are vierbein, spin connection, and

Tensor α is to be composed of O. We may rep-

an internal metric in the tangent space. The correspond-

resent the above term in the action also through

ing action contains new terms, which depend on these

the tetrad components of the derivatives of O:

fields. In general case the signature of the metric is de-

E^µcD_µOab = Oab;c. Modified Einstein-Cartan action

∫

√

termined dynamically. The Minkowski signature is pre-

reads S_ω = -m^2P

d⁴xe

OR^aµνbE^µaE^νdO^bd. Here R is

ferred dynamically because the configurations with the

curvature of gauge field ω. Besides, we may consider

other signatures are dynamically suppressed. We also

terms quadratic in curvature. In order to classify these

discuss briefly the motion of particles in the background

terms we introduce first the tetrad components of

of the modified black hole configuration, in which inside

curvature: R_abcd = E^µcE^νd O_adR^dµνb. The general form

the horizon the signature is that of Euclidean space-

of the action quadratic in curvature has the form:

∫

√

time.

S_R

d⁴xe

ORa1b1c1d1 Ra2b2c2d2 γa1b¹c¹d¹a²b²c²d² ,

Our first basic variable is vierbein e^aµ, which is

tensor γ is composed of matrices O. Another terms

matrix 4 × 4. Metric is composed of vierbein as fol-

in the action may be composed of the covariant

lows g_µν

= Oab eaµ e^bν, the real symmetric matrix O

derivative of vielbein e^aµ;ν

= D_νe^aµ. We define the

is our second dynamical variable, which plays the role

tetrad components of the derivatives of vierbein as

of metric on tangent space. The case of space-time

e_ab;c = O_adE^µbE^νcD_µe^dν. There may be several indepen-

with Minkowski signature corresponds to the choice

dent terms quadratic in this derivative. Those ones have

∫

√

O = diag(1,-1,-1,-1) while the case of Euclidean

the form S_e

d⁴xe

Oea1b1;c1 ea2b2;c2 ζa1b¹c¹a²b²c² .

signature is O

= diag(1, 1, 1, 1). The choices O

The most general form of tensor ζ is given

diag(-1, 1, 1, 1) and O

= diag(-1, -1, -1, -1) also

in our paper. There is also the mixed term

∫

√

represent Minkowski and Euclidean signatures corre-

S_Oe

d⁴xe

O ea1b1;c1Oa2b2;c2ηa1b¹c¹a²b²c² with

spondingly. The cases O = diag(-1, -1, 1, 1) and O =

parameters ηa1b¹c¹a²b²c² . Finally, one may add the

∫

√

= diag(1, 1, -1, -1) represent the signature, which is

trivial cosmological constant term: S_λ = -λ

d⁴xe

typically not considered in the framework of conven-

Partition function may be written as

tional quantum field theory. O(4) transformations Ω of

∫

vierbein e^aµ → Ω^abe^bµ together with rescaling e^aµ → Λ^abe^bµ

Z = DeDODωe-SO-Sω-SR-Se-SOe-Sλ.

(where Λ = diag(λ₁, λ₂, λ₃, λ₄) with positive λ_i) are

able to reduce the general form of matrix O to the six

One can always choose the coefficients in the action

above mentioned canonical forms.

(α_i, ζ_σ, γ_σ, η_σ, λ) in such a way that the action is

√

We introduce connection ω^aµb that belongs to algebra

bounded from below for the case of real positive

detO.

of SL(4, R). As well as in conventional case we define

Moreover, we require that Euclidean action is positively

the inverse vierbein matrices through E^µaeaν = δµν,

defined. This allows to define the self-consistent quan-

E^µae^bµ = δ^ba while metric with upper indices is defined

tum theory. In this theory the fluctuations of fields

through g^µν g_νρ

= E^µaE^νbO^abe^cνe^dρO_cd

= δ^µρ with O

appear to be exponentially suppressed for real posi-

√

matrix such that O^abO_bc

= δ^ac. One can construct

tive

detO. At the same time negative detO results

the following action quadratic in the derivatives of O:

in the appearance of imaginary unity in the exponent.

The corresponding configurations are not exponentially

¹⁾e-mail: mikhailzu@ariel.ac.il.

suppressed and dominate over the configurations with

Письма в ЖЭТФ том 116 вып. 1 - 2

2022

Riemann-Cartan gravity with dynamical signature

positive detO. This is the way how the signature

as a toy model of the black hole-like configuration with

(1, -1, -1, -1) (or (-1, 1, 1, 1)) is distinguished dynam-

the signature change. On the background of this config-

ically.

uration the motion of a massive particle is considered

As an illustration of our general construction we con-

briefly. It is worth mentioning that such a configura-

sidered roughly the modification of the black hole solu-

tion may appear at a certain stage of the gravitational

tion, in which outside of the horizon it looks like an ordi-

collapse, when the singularity appears in the classical

nary Schwarzshield solution (considered in Gullstrand-

solution of Einstein equations close to center of the BH.

Painleve reference frame). Inside of the horizon the ex-

Then the quantum dynamics comes into play, and the

pression for the vielbein remains the same as in the or-

signature change might occur inside the horizon.

dinary Painleve-Gullstrand black hole, but the matrix

This is an excerpt of the article

“Riemann-

O_ab changes signature to that of Euclidean space-time.

Cartan gravity with dynamical signature”. Full text

We do not have an intention to consider the given config-

of the paper is published in JETP Letters journal.

uration as a real classical solution, buth rather look at it

DOI: 10.1134/S0021364022601002

Письма в ЖЭТФ том 116 вып. 1 - 2

2022