R=0 wormholes satisfying energy conditions in scalar ... · We obtained R = 0 wormholes in scalar-tensor gravity. ds2 = − " C1 m + β r + C2 r 1 − 2m r − β r2 #2 dt2 + dr2

R = 0 wormholes satisfying energy conditions in scalar-tensorgravity

Rajibul Shaikh

Centre for Theoretical StudiesIndian Institute of Technology Kharagpur, India

Work done in collaboration with

Prof. Sayan KarDept. of Physics and Centre for Theoretical Studies

Indian Institute of Technology Kharagpur, India

JGRG 2016Osaka City University, Osaka, Japan

October 24-28, 2016

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 1 / 13

Plan of the talk

Wormholes and energy conditions

R = 0 wormhole solution in scalar-tensor gravity

R = 0 wormholes and the energy conditions

Gravitational lensing by the R = 0 spacetimes

Summary

References

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 2 / 13

Wormholes and energy conditionsA topological or geometrical short cut or tunnel between two universes or twoseparate regions of the same universe.

The spacetime geometry (M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988))

ds2 = −e2Φ(r)dt2 +dr2

1− b(r)r

+ r2(dθ2 + sin2 θdφ2)

Φ(r) → redshift functionb(r) → shape function

Wormhole throat at r = r0 such that

b(r0) = r0

and Φ(r) is finite in r0 ≤ r < ∞ (no horizon condition).

t = constant and θ = π/2 section is embedded in background cylindricalcoordinates (z, r, φ) system

ds22 = dz(r)2 + dr2 + r2dφ2

=

[

1 +

(

dz

dr

)2]

dr2 + r2dφ2.

z(r) → embedding function

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 3 / 13

Wormholes and energy conditions

Comparing with the wormhole metric

dz

dr= ±

√

b/r

1− b/r.

Flare-out condition at the throat

d

dz

(

dr

dz

)

=b− b′r

2b2> 0

Raychaudhuri equation

dθ

dλ+

1

2θ2 + σ2 − ω2 +Rαβ u

αuβ = 0,

Expansion: θ = ∇iui = ± 2

re−Φ

√

1− b(r)r

θ(r0) = 0 and dθdλ

> 0 on both sides. Alsoσ2 = 0, ω2 = 0. Therefore, around the throat

Rαβ uαuβ < 0

⇒ Violation of convergence condition.

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 4 / 13

Wormholes and energy conditions

In general relativity (GR)

Rαβ uαuβ < 0 ⇒ Tαβu

αuβ < 0

⇒ Violation of null energy condition.⇒ Violation of other (weak, strong, etc.) energy conditions.

But, in scalar-tensor gravity

Gµν =κ

φTmattµν +

κ

φT φµν

In such theoryRαβ u

αuβ < 0 ; Tαβuαuβ < 0

⇒ We may have wormholes with matter satisfying energy conditions.

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 5 / 13

Scalar-tensor gravity

The action

S =1

2κ

∫

d4x√−g

[

ΦR − ω(Φ)

Φ(∂σφ)

2

]

+ SM (g,Ψ)

The field equations

Gµν =κ

ΦTMµν +

1

Φ(∇µ∇νΦ− gµν∇α∇αΦ)+

ω(Φ)

Φ2

(

∇µΦ∇νΦ− 1

2gµν∇αΦ∇αΦ

)

∇α∇αΦ = κTM

2ω(Φ) + 3− 1

2ω(Φ) + 3

dω

dΦ∇αΦ∇αΦ

ω = constant ⇒ Brans-Dicke gravity. We consider

ω(Φ) = − 3Φ

2(1 + Φ)

Scalar-tensor gravity with the above form of ω(Φ) also arises as a effectiveon-brane gravity (S. Kanno and J. Soda, PRD 66, 083506 (2002), T. Shiromizu, K.Koyama, PRD 67, 084022 (2003)) in the context of the two-brane RandallSundrum model. Φ measures inter-brane distance.

R = κTM ⇒ trace-less matter implies R = 0 and vice-versa.

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 6 / 13

R=0 wormholes

The metric

ds2 = −f2(r)dt2 +dr2

1− b(r)r

+ r2(

dθ2 + sin2 θdφ2)

R = 0 equation(

1− b

r

)

f ′′(r) +4r − 3b− b′r

2r2f ′(r)− b′

r2f(r) = 0

We choose the shape function to be b(r) = 2m+ β

r.

The general spacetime (R. Shaikh and S.Kar, PRD 94, 024011 (2016))

ds2 = −[

C1

(

m+β

r

)

+ C2

√

1− 2m

r− β

r2

]2

dt2 +dr2

1− 2mr

− β

r2

+ r2dΩ2

Depending on the parameter values, the metric represents black hole, wormholeor naked singularity.

Different speacial cases have been obtained mostly in brane-world gravity.

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 7 / 13

R=0 wormholes

Coordinate transformation: r =(

1 + mr+ m2+β

4r2

)

r

The metric in the isotropic coordinate becomes

ds2 = − h2(r)

U2(r)dt2 + U2(r)

(

dr2 + r2dΩ22

)

whereh(x) = (C1m− C2) (q1 + x) (q2 + x) , U(x) = 1 + x2 + 2µx

and

x =

√

m2 + β

2r, µ =

m√

m2 + β=

q1q2 + 1

q1 + q2, η =

C2

C1m=

q1q2 − 1

q1q2 + 1

For wormholes, we must have η > −1.

Defining ξ =√1 + Φ, the scalar field equation yields

ξ =2γ

q1 − q2log

∣

∣

∣

∣

q1 + x

q2 + x

∣

∣

∣

∣

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 8 / 13

R=0 wormholes and the energy conditions

The energy conditions are satisfied if

ρ ≥ 0, ρ+ τ ≥ 0, ρ+ p ≥ 0, ρ+ τ + 2p = 2ρ ≥ 0

In the limit x → 0 (i.e. r → ∞), we have

ρ = Q2x4 +O(x5), ρ+ τ = Px3 +O(x4), ρ+ p = −1

2Px3 +O(x4)

where

Q2 =16µ2

m2

[

(µ2 − 1)(ξ20 − 1) +γ2

q21q22

− γξ0q21q

22

(q1 + q2 − 2µq1q2)

]

P =8µ2

m2q1q2

[

−(ξ20 − 1)(q1 + q2) + 4γξ0]

We must set P = 0. This yields

γ2 =(q1 − q2)

2(q1 + q2)

4(q1 + q2)(logq1q2)2 − 8(log q1

q2)(q1 − q2)

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 9 / 13

R=0 wormholes and the energy conditionsAt large r (i.e. x → 0)

ρ =Q2

r4+O

(

1

r5

)

, τ = −Q2

r4+O

(

1

r5

)

, p =Q2

r4+O

(

1

r5

)

Assymptotically, the matter behaves like the Maxwell field.

m = 0 ⇒ q1 = − 1q2

= 1

1−√

1+δ2and Q2 = γ2

[

1− 2√1 + δ2 log

∣

∣

1+√

1+δ2

1−√

1+δ2

∣

∣

]

< 0

⇒ The mass term is necessary to satisfy the energy conditions

0 1 2 3 4 50

1

2

3

4

5

q1

q 2

(a)

0 0.00005 0.00010.

2. ´10- 13

4. ´10- 13

0.0 0.2 0.4 0.6 0.8 1.00

50

100

150

200

x

(b) Plot of ρ (blue), ρ + τ (red) and ρ + p (green).q1 = 1.5, q2 = 2.0

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 10 / 13

Gravitational lensing by the R = 0 wormholes(R. Shaikh and S. Kar, in preparation)

0.0 0.2 0.4 0.6 0.8-1

0

1

2

3

4

5

utp

DΦ

(c)

-5 0 5

-4

-2

0

2

4

6

r cosΦ

rsi

nΦ

(d) b = 1.5

Energy conditios satisfied if Q2 ≥ 0 (boxedregion).Always positive deflection angle if(q1 − q2)

2 > 4q21q22 (shaded region)

If the energy conditions are satisfied, (boxedregion), the deflection angle is alwayspositive (shaded region).

But, if the energy conditions are violated(outside boxed region), deflection may beeither positive or negative.

0 1 2 3 4 50

1

2

3

4

5

q1

q 2

(e)Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 11 / 13

Summary

We obtained R = 0 wormholes in scalar-tensor gravity.

ds2 = −[

C1

(

m+β

r

)

+ C2

√

1− 2m

r− β

r2

]2

dt2 +dr2

1− 2mr

− β

r2

+ r2dΩ2

Unlike in GR, the R = 0 wormholes satisfy the energy conditions for wide rangesof parameters.

The mass term m is necessary to satisfy the energy conditions by the R = 0wormholes.

If the energy conditions are satisfied, the deflection of light is always positive. But,it may be either positive or negative, if the energy conditions are violated.

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 12 / 13

References I

M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).

M. Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP, College Park, MD, 1995).

R. M. Wald General Relativity, (University of Chicago Press, Chicago, 1984).

C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961).

P. G. Bergmann, Int. J. Theor. Phys.1, 25 (1968).

Y. Fujii, K. Maeda, The Scalar-Tensor Theory of Gravitation (Cambridge University Press, Cambridge, England, 2003).

S. Kanno and J. Soda, Phys. Rev. D 66, 083506 (2002).

T. Shiromizu and K. Koyama, Phys. Rev. D 67, 084022 (2003).

R. Shaikh and S. Kar, Phys. Rev. D 94, 024011 (2016).

Thank You

Rajibul Shaikh (CTS, IIT Kharagpur) R = 0 Wormholes October 24-28, 2016 13 / 13