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  • From Here to Eternity and Back: Are Traversable Wormholes Possible?

    Mary Margaret McEachernwith advising fromDr. Russell L. HermanPhys. 495 Spring 2009April 24, 2009Dedicated in Memory of My Dear Friend and UNCW Alumnus Karen E. Gross (April 11, 1982~Jan. 4, 2009)

  • Do Physicists Care, and Why?

    Are wormholes possible?

    How can we model to prove or disprove?

    Wormholes are being taken seriously

    Possible uses?

    Interstellar travel

    Time travel

  • Outline

    History Models that fail Desirable traits General relativity primer Morris-Thorne wormhole Curvature Stress energy tensor Boundary conditions and embeddings Geodesics Examples

    Time machines and future research

  • What is a Wormhole?

    Hypothetical shortcut between distant points

    General relativity

  • Historical Perspective

    Albert Einstein general relativity ~ 1916

    Karl Schwarzschild first exact solution to field equations ~ 1916

    Unique spherically symmetric vacuum solution

    Ludwig Flamm ~ 1916

    White hole solution

    http://archive.ncsa.uiuc.edu/Cyberia/NumRel/Images/rel1916.jpg

  • Historical Perspective

    Einstein and Nathan Rosen ~ 1935

    Einstein-Rosen Bridge first mathematical proof

    Kurt Gdel ~ 1948

    Time tunnels possible?

    John Archibald Wheeler ~ 1950s

    Coined term, wormhole

    Michael Morris and Kip Thorne ~ 1988

    Most promising wormholes as tools to teach general relativity

  • Traversable Wormholes?

    Matter travels from one mouth to other through throat

    Never observed

    BUT proven valid solution to field equations of general relativity

  • Black Holes Not the Answer

    Tidal forces too strong

    Horizons

    One-way membranes

    Time slows to stop

    Schwarzschild wormholes

    Fail for same reasons

    http://images.google.com/imgres?imgurl=http://www.larrydsmith.com/astro/tidal.jpg&imgrefurl=http://www.larrydsmith.com/astro/blackholes.html&usg=__wNIsiFXRwehoFvdicnWzRLT6SJY=&h=283&w=356&sz=21&hl=en&start=35&um=1&tbnid=z5uEBCIEXF80xM:&tbnh=96&tbnw=121&prev=/images%3Fq%3Dtidal%2Bforces%2Bimages%26ndsp%3D20%26hl%3Den%26sa%3DN%26start%3D20%26um%3D1

  • Simple Models Geometric Traits

    Everywhere obeys Einsteins field equations

    Spherically symmetric, static metric

    Throat connecting asymptotically flat spacetime regions

    No event horizon

    Two-way travel

    Finite crossing time

  • Simple Models Physical Traits

    Small tidal forces

    Reasonable crossing time

    Reasonable stress-energy tensor

    Stable

    Assembly possible

  • A Little General Relativity Primer

    G GT 8

    Field equations relate spacetime curvature to matter and energy distribution

    Left side Curvature

    Right side Stress energy tensor

    Summation indices

    G R Rg 1

    2

  • More on General Relativity

    Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve. ~ Misner, Thorne, Wheeler, Gravitation

  • Spacetimes

    A spacetime is a four-dimensional manifold equipped with a Lorentzian metric[of signature](-,+,+,+). A spacetime is often referred to as having (3+1) dimensions. ~Matt Visser, Lorentzian Wormholes.

    http://2.bp.blogspot.com/_Vlvi--zU3NM/RzhJQ4gEThI/AAAAAAAAAE0/Wl8cadtTWho/s1600-h/babyinhand2.jpg

  • Representing Spacetimes: Flat

    Line Element

    Uses differentials

    Einstein summation

    Metric

    Minkowski space:

    ds g dx dx2

    ds g dt g dx g dy g dz

    g g g g cartesian

    g g g r g r spherical

    tt xx yy zz

    tt xx yy zz

    tt rr

    2 2 2 2 2

    2 2 2

    1 1 1 1

    1 1

    ; ; ; ( )

    ; ; ; sin ( )

  • Morris-Thorne Wormhole (1988)

    ds e dtb

    r

    dr r d d2 2 2 2 2 2 2 21

    1

    ( sin )

    (r) redshift function

    Change in frequency of electromagnetic radiation in gravitational field

    b(r) shape function

  • Properties of the Metric

    Spherically symmetric and static

    Radial coordinate r such that circumference of circle centered around throat given by 2r

    r decreases from + to b=b0 (minimum radius) at throat, then increases from b0 to +

    At throat exists coordinate singularity where r component diverges

    Proper radial distance l(r) runs from - to + and vice versa

    ds e dtb

    r

    dr r d d2 2 2 2 2 2 2 21

    1

    ( sin )

    -

    +

  • Morris-Thorne Metric

    g

    g

    g

    g

    g

    e

    b

    r

    r

    r

    tt

    rr

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    0 1 0 0

    0 0 0

    0 0 0

    2

    1

    2

    2 2

    sin

    ds e dtb

    r

    dr r d d2 2 2 2 2 2 2 21

    1

    ( sin )

  • Determining Curvature (Left Side)

    Do we have the desired shape?

    Cartans Structure Equations

    Elie Joseph Cartan(1869~1951)

    d

    d

    i

    j

    i j

    j

    i

    j

    i

    k

    i

    j

    k

    G GT 8

    Cartan I

    Cartan II

  • Cartan I Connection One-Forms

    One-forms

    Take from Morris-Thorne metric:,

    dx dy dz

    pdx qdy rdz

    j

    i

    i

    j

    , ,

    0

    1

    1

    2

    2

    3

    1

    e dt

    b

    rdr

    rd

    r d

    sin

    ds e dtb

    r

    dr r d d2 2 2 2 2 2 2 21

    1

    ( sin )

  • Operations with Forms

    Wedge product

    d Operator: k-form to (k+1)-form

    Combining:

    dt dt dr dr d d d d

    dt dr dr dt

    0

    1

    form

    a a t r dada

    dtdt

    da

    drdr( , )

    a t r dt dda

    dtdt dt

    da

    drdr dt

    da

    drdr dt( , )

  • Calculation Connection One-Forms

    0 0 0

    1

    1

    21

    1

    21

    2 2 2

    3 3 3

    1

    1 0 1

    1

    1

    e dt d e dt dr dte

    b

    rdr d dr

    b

    r

    rd d dr d dr

    r d d dr d r d d dr

    ;

    ;

    ;

    sin sin cos ;sin

    ds e dtb

    r

    dr r d d2 2 2 2 2 2 2 21

    1

    ( sin )

  • Calculation Connection One-Forms

    d e dt dr

    e dt dr eb

    rdt

    eb

    rdt

    eb

    rdt

    0

    1

    0 1

    2

    0 2

    3

    0 3

    1

    0 1

    1

    0 1

    1

    21

    1

    0

    1

    2

    0

    1

    1

    2

    1

    1

    1

    d

    i j t r

    i

    j

    i j

    , , , ,

  • Matrix of One-Forms

    j

    i

    eb

    rdt

    eb

    rdt

    b

    rd

    b

    rd

    b

    rd d

    b

    rd d

    0 1 0 0

    1 0 1 1

    0 1 0

    0 1 0

    1

    2

    1

    2

    1

    2

    1

    2

    1

    2

    1

    2

    sin

    cos

    sin cos

  • Calculation Curvature Two-Forms

    Computed using matrix of one-forms

    Non-zero components of Riemann tensor

    Useful for computing geodesics

    j

    i

    j

    i

    k

    i

    j

    k

    mnj

    i m n

    d

    R

    1

    2

    i j k m n t r, , , , , , ,

  • Calculation Curvature Two-Forms

    3

    2

    3

    2

    0

    2

    3

    0

    1

    2

    3

    1

    2 3

    3

    2 3

    3

    3

    1 1

    d

    b

    rd d

    b

    r r r

    b

    r

    R Rb

    r

    R Rb

    r

    sin

    sinsin

    3

    2

    3

    2

    cos

    sin

    d

    d d d 0

    2

    3

    0 0

    1

    2

    1

    2

    3

    1

    1

    2

    1

    1

    b

    rd

    b

    rdsin

    dr

    dr

    1

    1

    2

    3

    sin

  • Riemann Tensor Components

    R R R Rb

    r

    b r b

    rrtr

    t

    ttr

    r

    ttr

    r

    trt

    r

    1

    2

    2

    2

    R R R R

    b

    r

    rt

    t

    t t t

    t

    tt

    1

    R R R R

    b

    r

    rt

    t

    t t t

    t

    tt

    1

    R R R Rb r b

    rr

    r

    r r r

    r

    rr

    2 3

    R R R Rb r b

    rr

    r

    r r r

    r

    rr

    2 3

    R R R Rb

    r

    3

  • Riemann Tensor Properties

    Antisymmetric in (t, r) Antisymmetric in (,) Symmetric in (t, r) and (, ) Only 24 independent components Generally in 4D has 256 components Governs difference in acceleration of two

    freely falling particles near each other

    R R R Rtr rt tr tr

  • Ricci Curvature Tensor

    R R R R R

    b

    r

    b r b

    r

    b

    r

    r

    tt ttt

    t

    trt

    r

    t t t t

    12

    2 12

    2

    R R R R R

    b

    r

    b r b

    r

    b r b

    r

    rr rtr

    t

    rrr

    r

    r r r r

    12

    2

    2 3

    G R Rg 1

    2

  • Ricci Curvature Tensor

    R R R R R

    b

    r

    r

    b r b

    r

    b

    rR

    t

    t

    r

    r

    1

    2 3 3

    R R R R R

    b

    r

    r

    b r b

    r

    b

    rR

    t

    t

    r

    r

    1

    2 3 3

    G R Rg 1

    2

  • Curvature Scalar

    R R R R R

    b

    r

    b r b

    r

    b

    r

    r

    b r b

    r

    b

    r

    tt rr

    2 1

    4 12 22

    2 3 3