A Metric Theory of Gravity with Torsion in Extra-dimension

Kameshwar C. Wali(Syracuse University)

Miami 2013[Co-authors: Shankar K. Karthik and Anand

Balaraman]

• Introductory Remarks• Basic Formalism• A set of Constraints; Motivation• Torsion, Connection Coefficients in terms of

Metric• Einstein Equations• Modified Einstein Equations• Modified FRW equations• Spherically Symmetric Solutions• Conclusions

OUTLINE

Introductory Remarks• Metric gij and connection are two basic

constituents of the manifold structure of space-time.

• Metric compatibility leads to the determination of the connection,

• When torsion is zero, connection is determined by the metric and its derivatives. In general metric and torsion are two independent characteristics of Riemannian Geometry.

• Because of the immense success of GR with torsion vanishing, torsion is not generally considered in the usual Einstein theory of gravity

• In the present work, we consider a theory of gravity with a hidden extra-dimension and metric dependent torsion

4

4

1 1

1

BASIC FORMALISM: 5D Metric

Constraints and their implications

• The two Constraints:

• The first ensures that the torsion components in 4D are zero

• The second ensures that 5-D has no effect on 4D geodesics.

The above constraints are sufficient to uniquely determine the non-vanishing torsion components and connection coefficients in terms of the metric.

Connection Coefficients in terms of the Metric

Einstein Equations

The 4D metric does not depend on x^5!! The other metric components can depend upon x^5.

Ricci tensor and Ricci scalars are constructed from the 4D metric and hence at the kinematic level, the theory is exactly the same as GR. Geodesic motions are not affected

Action Principle leading to Modified Einstein Equations

Homogenous – Isotropic Cosmology

The 4D metric for such cosmolgy:

Using the standard Einstein tensor for this metric and the modified Einstein Eqns.,

• In deriving the above equations, we have assumed metric is independent of x^5, and H satisfies the Eqn.,

Also stress tensor to be perfect fluid. With P=0,

q0 is the current value of the deceleration parameter

Some General Comments

Figure shows the behavior of a(t) for various values of q0.

• For the flat space topology, the universe does not originate from a big bang singularity for all values of q0 < +0.5.

• For the spatially closed topology, we find oscillatory solutions for all q0 <+1.

•

Robertson-Walker CosolmogyHomogeneous-Isotropic metric

a

t

1

a

t

1

Closed space k=1Flat space k=0

amin amin

No big bang (amin>0) if qo < +0.5 No big bang (amin>0) if qo < +1.0

qo = - 0.5 amin= 0.56

qo = - 0.5 amin= 0.65

By increasing qo we can take amin arbitrarily close to 0, as small as the size of CMB scale or even nucleosynthesis scale.

Static Spherically Symmetric Vacuum Solutions

The 4D metric has the general form,

Which when combined with modified Einstein equations, it leads to the following coupled equations,

In deriving the above Equations, we have used the cylindrical condition, which leads to

and the additional terms to Einstein Eqns.,

Some general remarks

• If J(r) is a constant, it has to be identically zero.

• If J(r) vanishes at some point, it has to vanish identically everywhere. The simplest solution,

the well-known Schwarzschild solution

Solutions for the metricWith

Numerical Solutions

• Metric solutions

General Results

From the above behavior of F(r), we are led to the behavior of A(r) & B(r),

Qualitative behavior

• When M=[cλ-1], A(r) is a monotonically increasing function leading to attractive gravity, and it corresponds to a Schwarzschild solution at large r when c>>1.

• The solutions have no event horizon. Both A(r) and B(r) are finite and regular for r>0. At r=0,

there is a physical singularity, a naked singularity.

Summary and Conclusions• Metric and torsion are two independent constituents of metric compatible Riemannian

Geometry. Because of the immense successes of GR, torsion is generally neglected in theories of gravity. However, in unified theories of gravity and other interactions, a more general theory including torsion becomes imperative.

• In the present work, torsion is incorporated in a novel way in higher dimensional KK type theories. With a set of suitable constraints, torsion is determined in terms of the metric. Non-vanishing torsion components are are confined to the extra dimension, leaving the 4D space-time free of torsion. The resulting modified Einstein equations , however, have significant physicsl consequences.

• In Isotropic-Homogeneous cosmology, possibilities other than the big bang standard FRW cosmology. Acceleration parameter can be chosen as an independent condition. No such freedom in FRW cosmology.

• In the case of spherically symmetric vacuum solutions, positive mass solutions with naked singularity open up the possibility of an arbitrarily large star collapsing to an arbitrarily small non-singular state. Since trapped surfaces would not necesaarily form in such collapses, finite matter pressure may be sufficient to withstand the singularity.

Brief Summary

• Including torsion in higher dimensional theories, but confining it to the higher dimension leads to modified Einstein equations that have important and interesting physical consequences

• Homogeneous-isotropic cosmology, possibity of no big-bang singular universe; Oscillatory universe. Spherically symmetric vacuum solutions with naked singularity, with no event horizon.