16.TimeMachine

7/30/2019 16.TimeMachine

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MTY Traversible Wormhole

6. Time Travel 2

l of the following are time travel spacetimes. Whether or not some of the following are, in

dition, time machine spacetimes is still open to debate.

Morris, M. and K. Thorne (1988) Wormholes in spacetime and their use for interstellar travel,Am. J. Phys. 56, 395.Morris, M., K. Thorne, U. Yurtserver (1988) Wormholes, time machines and the weak energycondition, Phys. Rev. D 49, 3990.

call the Einstein-Rosen wormhole: It doesnt stay open long enough.

A worldline connecting two points in I and IV during the time the wormhole is open must at some point become spacelike (slope less that 45).

A B C

I

IV

I

IV

A

B

C

SK #1: To keep wormhole throat open long enough for use by timelike travelers.

Intuitively, want to strech maximally extended Schwarzschild spacetime so that is now everywhere timelike. s t

r e t c

h !

OW?: Requires negative energy! In fact, negative energy allows awormhole solution to the Einstein equations without event horizons!

UT: In what sense is such a traversible wormhole a time machine?

A time travel spacetime is a solutiono Einsteins equations that admitslosed timelike curves ( CTCs ).

A time machine spacetime is a time travelspacetime in which the CTCs are generated bythe operation of a device (the time machine).


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time

space

t = 1005

t = 2005A

B

General wormhole : Mouths A and B are separated by a spatial and a temporal distance. If their spatial distance is so great that it would take a (timelike) traveler longer to get from one to the other than their temporal distance, then the wormhole shouldnt count as a time machine.

Suppose mouth A is on Earth and mouth B is in a galaxy 2000 light-years away. If you

jump through A, you go back in time 1000 years, but it would take you more than 2000 years to get back to the Earth!

t = 1005

t = 2005A B

Temporally-localized wormhole : Mouths are separated in space only. A convenient way to travel, but not a time machine!

Spatially-localized wormhole :Mouths are separated in time only.At last a time machine!

t = 1005

t = 2005A

B

ow to construct a time machine from a temporally localized wormhole:

1. Take one mouth on a twin-paradox trip ( special relativistic time dilation ).

2. Place one mouth closer to an intense gravitational source ( general relativistic gravitational red-shift ).

OR


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oblem: Radiation feedback : Light entering A exits B in the past; re-enters A again withformer self; re-enters A again with former selves... etc . Feedback becomes infinitealmost instantaneously and BOOM !

e solution: Roman Configuration = Two or more traversiblewormholes, each one of which doesnt count as a timemachine, but entire configuration does.

t = 1005

t = 2005A

B

C D

A and B are far enough apart spatially to prevent radiation

feedback. Use (temporally-localized)wormhole CD to get back to spatial location of A in the past.

Problems:1. Negative-energy exotic matter leads to quantum mechanical instabilities.2. Availability?

T. Roman, T. (1993)Phys. Rev. D 47, 1370

usal structure of wormhole time machine:

time

space

mouth Amouth B

Cauchy horizon

chronology-violating region

Wormhole time-machine constructed via twin-paradoxtime dilation.

The chronology-violating region is the interior of the lightcone generated by the null surface .

One Attempted Solution: Use ringhole wormholes: mouths are tori, not spheres (Gonzales-Diaz, P.1996 Phys. Rev. D 54, 6122).

Note : This diagram doesn't show the wormhole "throat" which connects its mouths: it just shows the worldtubes of the mouths.


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Ori-Soen SpacetimeOri, A. (1993) Must time-machine construction violate the weak energy condition?, Phys. Rev. Lett . 71, 251Ori, A. and Y. Soen (1994) Causality violation and the weak energy condition, Phys. Rev. D 49, 3990.

rely geometrical construction: Spacetime with a region shaped like a torus (doughnut). CTC svelop inside torus to the future of a spacelike hypersurface.

osed spacelike curve S

side torus: after a given me t 0 , this and others ar it become timelike.

S

S

Size of cross-section (shaded region)described by coordinate. The curve S goes through the point = 0.

aracteristics

No exotic matter required.Spacetime is flat far from torus and topologicallytrivial (no fishy identification of points/regions, etc ).Partial Cauchy surfaces exist.Future Cauchy horizon is compactly generated.

blems

Partial Cauchy surfaces are non-trivially curved.What matter distribution corresponds to thisgeometry and is it physically possible to construct?

= 0

H chronology-violating region

t = t 0N

t

e

t 0

Before t 0, the closed curve S through = 0 is spacelike.At t 0, the closed curve N through = 0 is lightlike.After t 0, the closed curve T through = 0 is timelike.

S

N

T

S

N

T

H

Causal Structure: is a nonflat spacelike partial Cauchy surface. The trough-like surface H is the future Cauchy horizon of . It extends upward from N (at t = t 0 ) and remains inside the interior of the torus. The chronology-violating region is inside the trough.

A cut-away view of the causal structure.All t = const. surfaces before t = t 0 are spacelike and qualify as partial Cauchy surfaces. (After t = t 0 , all t = const.surfaces no longer are purely spacelike.)


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nse-Thirring Effect Time-Travel Spacetimes (Gdel, Tipler, Gott):

ese are spacetimes in which rotating matter induces a dragging of inertial frames (Lense-irring Effect) that creates CTCs.

Lense-Thirring Effect

Occurs when a massive rotating object drags spacetime in its vicinity (but veryimperceptivity for normal matter)

Rotating Earth: like a bowling ball in molasses...

For gyroscope in orbit around Earth, Lense-Thirring Effect causesaxis of spin to go out of alignment by an angle of 42 milli-arcseconds per year (width of a human hair from 1 km away!)

Gyroscope in orbit: gets dragged along with the molasses (but very imperceptivity)


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Gdel spacetime

time

space take out slice (not fully spacelike)

aracteristics

T is an accelerated worldline (not a geodesic). Requires rocket with huge amount of fuel!Spacetime is static and rotationally symmetric about each matter worldline (all observers see what A sees;all see matter spinning about them).

Gdel, K. (1949) An example of a new type of cosmological solution of Einsteins equations of gravitation, ReMod. Phys. 21, 477.

ghtcones tip over and flatten as we proceed away from A

ng concentric circles. Initially,h circles are closed spacelike ves S. Eventually lightcones to create closed null curves . and then tip further to create sed timelike curves T.

blems

No partial Cauchy surfaces exist, so no way to describe CTC s as due to a programmable time machine.Not physically relevant (our universe is expanding and non-rotating).

sees matter rotating about her

ns-Thirring Effect: Think of spacetime as being dragged und with the rotating erver (like molassas) to the ent that CTCs form.

A

T

N S

Accelerated trajectory of timelike observer accessing CTC by spiraling outward

S

N

T


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Tipler Cylinder Tipler, F. (1974) Rotating cylinders and the possibility of global causality violation, Phys. Rev. D 9, 2203.

acetime of an infinitely long rotating cylinder. The Lens-Thirring Effect drags region surrounding cylinder to thetent that CTC s form.

htcones straighten up as proceed away from nder along concentric les. Initially, such circles closed timelike curves T.ntually lightcones ighten up to create closed curves N... and then

sed spacelike curves S, S'.S

N T

S'

aracteristics

Centrifugal forces are balanced by gravitational attraction.Spacetime is asymptotically well-behaved.

blems t physically relevant (an infinite cylinder). Although can make adjustments.

Accelerated trajectory of timelike observer accessing CTC by spiraling inward

S'

N

T

exact solution for finite massless cylinder (Bonner, W. B. 2002, Clas Quant Grav 19, 5951)

exact solution for infinite light cylinder (Mallett, R. 2003 Found Phys 33, 130although some problems: Olum, K. &A. Everett 2004 arXiv: gr-qc/0410078)

There are closed timelike geodesics as well (Steadman, B. R.2003 Gen Rel and Grav 35, 1721)


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Gott Cosmic String spacetime Gott, J. R. (1991) Closed timelike curves produced by pairs of movingcosmic strings: Exact solutions, Physical Review Letters 61, 1446.

Cosmic string = thin strand of extremely dense matter

10 million billion tons per centimeter millions of light-years long non-zero width (very narrow) no endpoints (so either infinitely long or loopy)

ASIDE : Cosmic strings are 1-dimensional topological defects from phase transitions in the early universe,according to some cosmological models.Such defects occur at the borders of

regions that undergo differing phase transitions (0-dim defects are magnetic monopoles and 2-dim and higher defects are domain walls). Cosmic strings are not to be confused with superstrings from string theory. The latter are 0-width loops that are supposed to be the

fundamental constituents of matter.

Spacetime in region of a cosmic string: conical

string (cross-section)

identify

space

C

circumference of circle C < 2 radius

For string density = 10 million billion tons/cm, slicesubtends angle of 3.8 sec of arc (very small but noticable).

Minkowski spacetime with a wedge cut out


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me travel using cosmic strings:

Paths 1, 2, 3 are all geodesics. Paths 1 and 3 are shorter than path 2.

AB

pa t h 1 p a t h 1

path 2

pa t h 3 p a t h 3

string 2

string 1

identify

identify

region I

region II

space

us: From the point of view of an inertial observer moving

along path 2, the rocket is traveling faster than light !

w: Transform to observer C's rest-frame

For trip from A to B : Take region I and set it in motion to rightat constant speed v .For trip from B to A: Take region II and set it in motion to leftat constant speed v .

A rocket traveling at a sufficient constant speed (very close to c ) on path 1 can beat a lightsignal traveling on path 2.

uivalent to: A spacetime that satisfies the Einstein equations in which there are two cosmic

ings moving past each other at constant relative velocity v . CTCs (path 1 + path 3) form ineir near vicinity!

ASIDE: No attraction between strings due to negative energy string tension.

ASIDE: Each region is a separate complete solution to the Einstein equations, and they can be patched together smoothly to

form a single solution. Moreover,they can be patched together smoothly to form a solution in which there is a non-zero constant relative velocity v between them.

al: To construct a spacetime with CTCs .tial Set-up: Spacetime with 2 parallel cosmic strings

There is a particular inertial observer C moving at constant velocity v along path

2 for whom the departure of the rocket from A occurs after its arrival at B !

From the rocket's point of view, it never exceeds c.

Recall the discussionon tachyons.

d: The same holds for the rocket's return from B back to A along path 3.


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hat this construction describes:

wo cosmic strings parallel to each other moving in opposite directions as they pass by.acetime surrounding them admits CTCs !

Relative angular momentum between strings creates Lens-Thirring effect -drags timelike curve consiting of paths 1 and 3 into a closed loop.

string 1string 2

AB

v /2v /2

CTC

space

Chronology-violating region is the exterior region of the hour-glassCauchy horizon.

AB

worldline of a point on string 1

worldline of a point on string 2

space

time

Cauchy horizon

chronology-violating region chronology-violating

region

no time-travel

no time-travel

rocket worldline (CTC)

usal structure of Gott Cosmic String spacetime: Cutler (1992) Phys. Rev. D 45, 487

oblem: Does it count as a time machine? The Cauchy horizon is not compactly generated

(its generators come in from past infinity). So there isnt a sense in which you can

throw a switch (on some Cauchy surface ) and turn on a time-machine.

vantages:

i) No need for exotic matter.ii) Chronology-violating region exists for finite period of time -- possible reason why we havent

been invaded by hordes of tourists from the future (Hawking 92).

Rocket worldline describes rocket at rest at A for some period of time. Then follows CTC to B and back to A, arriving in past, then continues at rest at A.

Only a point on both strings is represented: the world-"sheets" of the strings stretch to spatial infinity.

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16.TimeMachine