Cosmology Midterm

7/24/2019 Cosmology Midterm

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Wormholes in spacetimeand their use for

interstellar travelMichael S. Morris and Kip S. Thorne

17 July 1987


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Basic wormhole criteria

Metric should be spherically symmetric and static.

(simplify the calculations)

Solutions obeys Einstein field equation

Solution must have a throat (tunnel) that connectstwo asymptotically flat regions

Traversable Wormholes


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Embedding Diagram


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Form of the Metric

Basic criteria requires the metric of the form

The Riemann, Ricci, and Einstein tensors

Viebien the proper reference frame of static

observer

ds2 = e2(r)dt2 + dr2

1 b(r)/r+r2d2

g(x) = e

(x)e(x)


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Einstein Field Equations

Stress-energy tensor

The field equations

Ttt = c2 Trr = T = T = p

=b0

8Gc2r2

= b/r 2(r b)0

8Gc4r2

p= r

2[(c2 )0 0]


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Other Constraints

No horizon, so as to allow two-way travel

There exists a minimum radius in the wormhole

The spacetime is asymptotically flat

Finite proper length

g00 e2 0

r = b = b0

b/r 0 and 0 as r

1 b/r 0


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Trouble Aspects

Field equations plus the constraints leads to

The mass-energy density of "exotic matter" as seen

by the travelers frame is negative!

0 > 0c2

T00

= 2(0c2 0) + 0


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Usability Criteria

Tidal gravitational forces experienced by a traveler

must be bearably small

The proper time of a traveler and the static observeroutside the wormhole should be small

The solutions should be perturbatively stable


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Wormholes, TimeMachine, and the Weak

Energy ConditionMichael S. Morris, Kip S. Thorne and Ulvi Yurtsever

26 Sept 1988


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Energy Condition

Any traversable wormhole a two-sphere

surrounding one mouth, the wormholes stress-energy condition must violate averaged null energy

condition (ANEC)ZC

Tkkd 0


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Time Machine


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Form of the Metric

The metric inside the accelerating wormhole

The metric guarantees that with tiny fractionalchanges of the stress-energy tensor, the

wormholes size and shape are held nearly constant

throughout the acceleration

ds2 = (1 +g(t)lF(l)cos )2 exp2 dt2 + 1

1 b(r)/rdr2 +r2(d2 + sin2 d2)


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Classical Instability

A high-frequency electromagnetic wave packet

moving along null-like world line gets blue shifted bya factor

Wormholes ANEC-induced action causes the wave

packet to spread laterally, amplitude down by afactor

The horizons are classically stable if

=p

(1 +)/(1

)

b/2

D

(b/2D)


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Closed Timelike

Curves

Kip S. ThorneFeb 1993


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A Though Experiments

What constraints do the laws of physics place on

the activities of an arbitrarily advanced civilization?

Do the law of physics prevent arbitrarily advancedcivilizations from constructing "time machine", and if

so, by what physical mechanism are theyprevented?

Hawking (1992) Chronology ProtectionConjecture


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Van Stockums Spacetime (1937)


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Chronology Protection Mechanism

Enforcement of ANEC

Klinkhammer, Gunnar. "Averaged energy conditions for free scalar fields in flatspacetime." Physical Review D 43.8 (1991): 2542.

Yurtsever, Ulvi. "Does quantum field theory enforce the averaged weak energycondition?." Classical and Quantum Gravity 7.11 (1990): L251.

Urban, Douglas, and Ken D. Olum. "Averaged null energy condition violation in aconformally flat spacetime." Physical Review D 81.2 (2010): 024039.APA

Not very promising

ZC

Tkkd 0


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Classically Instability wormhole is a

counterexample

Quantum-field instability: piling up of the vacuumfluctuations. This pile up causes the fluctuations to

have nonzero renormalized energy density thatdiverges as one approaches the horizon.

T = X

n=1

1/2n

62l2Pl3n

(2knl

n+ 2lnk

n+knk

n+lnl

n+glnk

n).


19/19


Quantum gravity Even if chronology is protected

macroscopically, quantum gravity, that is not yetwell understood, may well give finite probability

amplitudes for microscopic spacetime with CTCs.

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Cosmology Midterm

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