traversable wormhole solution

7/28/2019 Traversable Wormhole Solution

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TRAVERSABLE SCHWARZSCHILDWORMHOLE SOLUTION

Nur Izzati Ishak (SES100227)

Prof. Dr. Wan Ahmad TajuddinBin Wan Abdullah

Theoretical Physics Laboratory

Department of Physics, Facultyof Science, University Malaya

50603 Kuala Lumpur,Malaysia


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OUTLINESObjective

Introductiona)Einstein Field Equation

b)solution of Einstein field equation

Methodology

a)structural equation of wormhole

b)solving field equation-mathematics of embedding

-Exotic material

Results and Analysis

Energy density

Discussiona)Energy condition

b)Violation

Conclusions

References


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1.construct a traversable wormhole solution using

Morris-Thorne framework: spherically symmetric and static metric

In every point on the spacetime, the metric must fulfill

Einstein field equation.

No event horizon.

The solution must be consist shape of wormhole; throat

and flare-out condition

Sources of curvature for wormhole must have physicallyacceptable energy.

2. To derive the equation that describes the

exoticity of matter used as the source to generate

spacetime curvature.

OBJECTIVES


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INTRODUCTIONS


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Einstein field equation represent curvature

of space and matter.

Matter is characterized by energy

momentum tensor Its geometrized the gravitation

4

8 G

G Tc


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SOLUTION OF FIELD EQUATION

Schwarzschild black hole

2r m

1

2 2 2 2 2 2 22 21 1 sinm m

ds dt dr r d d r r

Singularity

(Spacetime shrink to zero volume V =0)


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SPACETIME METRIC

=Gravitational redshift

b=shape function

metric tensors(gravitational potential)

2200

ceg

rbg /111

222 rg

2233 sin rg

33221100

,,, ggggdiagg

rbg

/1

111

22

00ceg

r

2

22rg

22

33sinrg

gg /1

2222

2

22)(22sin

/1 ddr

rb

drdtceds

r

4320 ,,, xxrxtx


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CHRISTOFFEL SYMBOLS

Nine nonzero terms

From spacetime metric, derived Christoffel

symbols using:

, g

gggg

2

,


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'0

01

r

bce 1'

221

00

rb

br

b

2

'1

11

rb 122

21

33 sinrb

cossin233

r

1313

2

12

1

r

3

23cot


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RIEMANN AND RICCI TENSOR AND

RICCI SCALAR

Riemann tensors

Riemann tensors describe the curvature of wormholespacetime

Contraction of Riemann tensor yield Ricci tensor

the dif ferenc e in accelerat ionof two freely falling

neighboring particles into the wormhole

R

R


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Ricci tensor

rbrr

brb

r

bceR

'2

2

'''''1

222

00

2

11 '"2

'2

'

rbrr

brbR

'2

''22 b

brr

brbrbR

22233 sinRR

Contraction of Ricci tensor yield Ricci scalar

33

33

22

22

11

11

00

00 RgRgRgRgRgR

r

b

r

brb

r

rb

r

rbR

'2

'''4'''

222

2


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EINSTEIN TENSORS

Represent GRAVITY

RgRG

21

2

22

00

'

r

bceG

rbr

b

rrbr

rbbG

''2'211

222

'"2

'''

2

'

2

rbr

brbbrb

r

bG

23333

sinGG


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EINSTEIN FIELD EQUATIONS

Orthonormal basis vector to represent

proper reference frame of observer who

remains at rest in coordinate system

with fixed

, , ,ct r

, ,r

Basis vectors:

eeeeeeeert

~,~,~,~~,~,~,~3210

Transformation

ee ~~

gg

gR

GG

Lemos et al(2003) Columbia University

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sin,,/1,

rrrbediag

0r is radius of throat a Is wormhole mouth


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STRESS ENERGY MOMENTUM TENSORS

Einstein Field equation via the Bianchi identity:

Nonzero components stress-energy tensors:

is energy density is radial tension

is tangential pressure

4

8T

c

GG

2

00)( crT )(

11rT )(3322 rpTT

)(r

)(r

)(rp


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SPACETIME METRIC PROFILE OF STRESS ENERGY COMPONENTS


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FIELD EQUATIONS

Total energy density

Radial tension

Tangential pressure

rr

b

r

b

G

cr

'128 3

4

r

r

br

brb

r

br

brbrb

Gcrp

12

''

12

''''18

)(22

2

2

2

2

'

8 r

b

G

c

r

'

2( 2c

rrp


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BIRKHOFF THEOREM

Existance of spherical (nontraversable)

Scwarzchild wormhole as the only solution that

can exist in vacuum Einstein field equation

Consequence??

Traversable wormhole must be associated with

matter or field wih nonzero stress energy tensor


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SOLVING THE EINSTEIN FIELD EQUATION

Firstly generate suitable geometry for wormhole solution by

controlling shape function and redshift function

Determine suitable stress-energy tensors from geometry

through the fields equation of andp,


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MATHEMATICS OF EMBEDDING

Function: To visualize the wormhole shape and obtain the

information on our chosen shape function.

An equatorial slice of static spherical symmetry geometry at

=

2is considered.

Re- parameterized using :

the representation of this line elements:

22222

22)(22

sin

/1

ddr

rb

drdtceds

r

22 2 2

1

drds r d

b r


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MATHEMATICS OF EMBEDDING

To visualize this line element ,embed this slice into to the 3-D Euclidean

space

metric in cylindrical coordinate is

Embedded surface is axially symmetric,

Minimum radius,

strict minimality at throat, the flare out condition:

zr ,,

22222

drdrdzds

)(rzz

2

1

1

rb

r

dr

dz

2

1

1

)(

rb

r

dz

dr

02

'22

2

b

rbb

dz

rd


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02

'22

2

b

rbb

dz

rd

2

1

1

rb

r

dr

dz

0z

a

r

rrb

drrl

0

2/1/)(1

)(

0z

a

rrrb

drrl

0

2/1

/)(1

)(


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Absence of even horizon:

is everywhere finite and as 0

l


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EXOTIC MATTER

Configuration needed to support wormhole:

Flaring out condition:

At wormhole throat , finite with and

Define theexoticityof material!!

0rbr 'b

2

2

c

c

'

'12

'

22

22

br

br

dz

rd

br

b

02

0

2

00

0

c

c

0'/1 rb


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Constraint on exoticity function:

the radial tension at throat is large as it must

surpassed the magnitude of the energy density

Property:

material exhibit this property known as exotic

matter What is means by this property???

2

00c

02 c


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RESULTS AND ANALYSIS


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ENERGY DENSITY

What is observation of energy density measured by statics

observers time?

Vectors static observers

Stress-energy tensor with basis vector on the observers

time

where

3210

~,~,~,~ eeee

10'0

~)/(~~ ecvee

11

22

01

2

00

2

'0'0)/(/ TcvTcvTT

0

22

0

2 )/( cvc

00

2

0

2 c 2/1

22 /1

cv


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As the observer move with very fast radial velocity (very large) he will observed negative energy

density

the matter in this region exhibit gravitational

repulsionproperties repulsion required negative energy densitywhich

only possible if the null energy condition is violated

Amount of exotic matter used in wormhole solutionis quantified via the 22)( ccr


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DISCUSSION


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ENERGY CONDITION

Can negative energy exist?

AWEC:

Average Point Wise Energy Condition along

the observers geodesic

as long as the average sum of this energy

density is positive.


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By Quantum Inequalities (QI)

inertial observer at Minkowski spacetime will

observed the negative energy in short period

of time

How about in curve space??

consider small spacetime volume at the

throat

the spacetime can be approximated as flat

surface throat is slightly larger than Planck

size


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VIOLATION

Hawking evaporation squeezed vacuum state in non -linear

optics


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CONCLUSIONS

considering a spherical symmetry spacetime metricwe obtained important equation of spacetimestructure in solving Einstein field equation .

Following Morris and Thorne approach in solving

these equations yield the definition of dimensionlessfunction known as exoticity function

This implies troublesome constraints on properties ofmaterial at the throat of wormhole where it must

exhibit negative energy density. The exoticity of thismaterial represents the measure of energy conditionsviolation.


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REFERENCES

1] C. W. Misner, K. S. Thorne & J. A. Wheeler. (1973). Gravitation. New York: W.H.Freeman andCompany.

[2] A.Einstein & N.Rosen. (1935). The particle problem in the general relativity. Physics Review,73-77.

[3]M.S.Morris & K.S.Thorne. (1988). Wormholes in spacetime and their use for interstellar travel:A tool for teaching General Relativity.America Journal of Physics, 5(56), 395-412.

[4] Lemos, J.P.S., Lobo, F.S.N. & Oliveira, S.Q. (2003). Morris-Thorne wormholes with acosmological constant. Physics Review D, 6(68), 064004

[5]A D.Benedictis & A Das (2001). On a general class of wormhole geometries. Classical andQuantum Gravity, 18(7), 181187

[6]M. Visser (1989). Traversable wormholes: Some simple examples. Physics Review D, 39(10),3182

[7] M. Safanova, D.F. Torres &G.E. Romero, (2006).Gravitational lensing by wormholes. PhysicsReview D, 74(2), 024020

[8] Frank J. Tipler, (1978). Energy conditions and spacetime singularities. Physics Review D,17(10), 25212528.

[9] L.H Ford & T.A Roman (1995). Averaged Energy Conditions and QuantumInequalities .Physics ReviewD ,51(8) ,4277-4286.

[10]M.Visser.(1995).Lorentzian Wormholes: From Einstein to Hawking. New York: AmericanInstitute of Physics
http://publish.aps.org/search/field/author/Frank%20J.%20Tiplerhttp://publish.aps.org/search/field/author/Frank%20J.%20Tiplerhttp://publish.aps.org/search/field/author/Frank%20J.%20Tipler

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