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Traversable Wormholes Possible? From Here to Eternity and Back: Are Traversable Wormholes Possible? Mary Margaret McEachern with advising from Dr. Russell L. Herman Phys. 495 Spring 2009 April 24, 2009 Dedicated 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 Historical Perspective Einstein and Nathan Rosen ~ 1935 “Einstein-Rosen Bridge” – first mathematical proof Kurt Gödel ~ 1948 Time tunnels possible? John Archibald Wheeler ~ 1950’s 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 Simple Models – Geometric Traits Everywhere obeys Einstein’s 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 8 GT 1 G R Rg 2 Field equations – relate spacetime curvature to matter and energy distribution Left side – Curvature Right side – Stress energy tensor Summation indices 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. Representing Spacetimes: Flat “Line Element” 2 Uses differentials ds g dx dx Einstein summation Metric Minkowski space: 2 2 2 2 2 ds gtt dt gxx dx g yy dy gzz dz gtt 1; gxx 1; g yy 1; gzz 1 (cartesian) 2 2 2 gtt 1; grr 1; g r ; g r sin (spherical) Morris-Thorne Wormhole (1988) 1 ds2 e2 dt 2 dr 2 r 2 (d 2 sin2 d 2 ) b 1 r Φ(r) – redshift function Change in frequency of electromagnetic radiation in gravitational field b(r) – shape function +∞ Properties of the Metric 1 ds2 e2 dt 2 dr 2 r 2 (d 2 sin2 d 2 ) b 1 -∞ r Spherically symmetric and static Radial coordinate r such that circumference of circle centered around throat given by 2πr 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 Morris-Thorne Metric 1 ds2 e2 dt 2 dr 2 r 2 (d 2 sin2 d 2 ) b 1 r e2 0 0 0 gtt 0 0 0 1 b 0 g 0 0 0 1 0 0 rr g r 0 0 g 0 2 0 0 r 0 0 0 0 g 2 2 0 0 0 r sin Determining Curvature (Left Side) Do we have the desired shape? G 8 GT Cartan’s Structure Equations “Cartan I” i i j d j “Cartan II” i i i k j d j k j Elie Joseph Cartan (1869~1951) Cartan I – Connection One-Forms “One-forms” dx, dy, dz 2 2 2 1 2 2 2 2 2 ds e dt pdrdx qrdy(drdz sin d ) b i j 1 j i r , Take from Morris-Thorne metric: 0 e dt 1 b 2 1 1 dr r 2 rd 3 r sind Operations with Forms “Wedge product” dt dt dr dr d d d d 0 dt dr dr dt “d” Operator: k-form to (k+1)-form 1 form da da a a(t,r) da dt dr dt dr Combining: da da da a(t,r)dt d dt dt dr dt dr dt dt dr dr Calculation – Connection One-Forms 1 ds2 e2 dt 2 dr 2 r 2 (d 2 sin2 d 2 ) b 1 1 0 e dt d 0 e dt r dr; dt 0 e 1 1 b 2 b 2 1 1 dr d1 0; dr 1 1 r r 1 2 rd d 2 dr d; d 2 r 1 3 r sind d 3 sindr d r cosd d; d 3 r sin Calculation – Connection One-Forms i i j d j i, j t,r,, 0 0 1 0 2 0 3 d 1 2 3 e dt dr 1 b 2 0 1 e dt dr 0 1 e 1 dt 1 1 1 r 1 2 0 b e 1 dt 1 r 1 b 2 1 e 1 dt 0 r Matrix of One-Forms 1 b 2 0 e 1 dt 0 0 r 1 1 1 2 2 2 b b b e 1 dt 0 1 d sin 1 d i r r r j 1 b 2 0 1 d 0 cosd r 1 b 2 0 sin1 d cosd 0 r Calculation – Curvature Two-Forms i i i k j d j k j i, j,k,m,n t,r,, 1 Ri m n 2 mnj Computed using matrix of one-forms Non-zero components of Riemann tensor Useful for computing geodesics Calculation – Curvature Two-Forms 2 2 2 0 2 1 2 3 d3 0 3 1 3 3 cosd 2 0 0 b d 2 sind d 0 3 sin d d 3 r 1 b 1 1 2 3 b 2 sin 2 r r r sin 1 1 d r b 2 3 1 r 3 2 1 b 3 sin1 d b R R 3 r r 1 2 b d R R r r 3 1 d 3 r sin Riemann Tensor Components b 2 br b Rt Rr Rr Rr 1 rtr ttr ttr trt r 2r 2 br b Rr R Rr R r rr r rr 2r 3 r r br b Rr Rrr Rr Rrr 3 b 2r 1 r R t R R t R t tt t tt r b 1 t t r R t Rtt Rt Rtt b r R R R R r 3 Riemann Tensor Properties Rtr Rrt Rtr Rtr Antisymmetric in (t, r) Antisymmetric in (θ,Φ) Symmetric in (t, r) and (θ, Φ) Only 24 independent components Generally in 4D has 256 components Governs difference in acceleration of two freely falling particles near each other Ricci Curvature Tensor t r Rtt Rttt Rtrt Rtt Rtt b 1 21 b 2 br b r G R1 Rg r 2 2r 2 r t r Rrr Rrtr Rrrr Rrr Rrr b 2 br b br b 1 r 2r 2 r 3 Ricci Curvature Tensor t r R Rt Rr R R b 11 G R Rg r br b b R 2 r 2r 3 r 3 t r R Rt Rr R R b 1 r br b b R r 2r 3 r 3 Curvature Scalar R Rtt Rrr R R1 G R Rg 2 b 41 b 2 br b r 2br b 2b 21 r r 2 r r 3 r 3 Einstein Tensor Einstein Tensor G Gtt ,Grr ,G ,G 1 G R Rg 2 “Ricci” Curvature Tensor R Rtt , Rrr , R , R Curvature Scalar R Rtt Rrr R R Metric g gtt , grr , g , g Einstein Tensor – Components 1 b G R Rg tt tt 2 tt r 2 1 1 b 2 b G RGrr Rrr RgRgrr 3 1 2 2 r r r 1 b br b 2 br b G R Rg 1 2 G 2 r 2r(r b) r 2r (r b) 1 b br b 2 br b G R Rg 1 2 G 2 r 2r(r b) r 2r (r b) Stress-Energy Tensor G 8 GT 0 0 0 0 0 0 T 0 0 p 0 0 0 0 p Equations of State Rearrange, solve…. b Energy density 8r 2 1 b Tension 2(r b) 8r 2 r r Pressure (stress) p ( ) 2 Wormhole Embedding Diagram 2 r r z(r) b ln 1, b 2 0 b b 0 0 0 Static (t=constant “slice”) Assume θ=π/2 (equatorial “slice”) Only r,Ф variable 1 b ds2 1 dr 2 r 2d 2 r ds2 dz 2 dr 2 r 2d 2 Boundary Conditions - Shape b0 – minimum radius at throat Vertical at throat Asymptotically flat r-axis Boundary Conditions – No Horizon Horizon - “physically nonsingular surface at g 1 whichds2 ett 2 vanishes”;dt 2 defineddr 2 r 2 (onlyd 2 forsin 2 d 2 ) spacetimes containing b one or more asymptotically flat1 regions r e.g. Schwarzschild metric – coordinate singularity at r=2M 1 2M 2M ds2 1 dt 2 1 dr 2 r 2 d 2 sin2 d2 r r Morris-Thorne metric e2 0 Other Boundary Conditions Crossing time on order of 1 year Acceleration and tidal acceleration on order of 1G Geodesics “Geodesics” are “extremal proper time worldlines”; equations of motion that determine geodesics comprise the “geodesic equation” ~ Hartle Timelike – Particle freefall paths; ds2 0 Null – Light freefall paths; ds2 0 LOCALLY Variational Principle 1 ds2 e2 dt 2 dr 2 r 2 (d 2 sin2 d 2 ) b 1 Lagrangian r 1 2 1 2 2 2 dx dx dt b dr d d 2 L g e2 1 r 2 sin2 d d d r d d d Euler-Lagrange Equation 1 d L L AB Ld 0 geodesics d dx x 0 d Geodesic Equation 2 1 d x dx dx g g , g , g , ; 2 d 2 d d ,, , t,r,, d dt t e2 0 d d 1 1 2 2 2 2 d b dr d b dr d d dt r 1 1 r sin2 e2 0 d r d dr r d d d d 2 d d d r 2 r 2 sin cos 0 d d d d d r 2 sin2 0 d d Christoffel Symbols Describe curvature in non-Euclidean space Metric is like first derivative of warp Christoffel symbol is like second derivative of warp Also called “connection coefficients” Non-zero Christoffel symbols are components of a 3-tensor t t d 1 r 2 rt tr (r b)sin dr r r r 1 r sin cos b r r r r r br b cos r b 2 1 e rr tt 2r(r b) sin r Morris-Thorne Examples Zero tidal forces Exotic matter limited to throat “Absurdly benign” wormhole Zero Tidal Force Solution 0 b b(r) bor r for b0 1 Equations of State 5 b 16r 2 8r 2 5 1 b 2(r b) 8r 2 8r 2 r 5 r p ( ) 2r 2 2 Zero Tidal Force Solution Wormhole material extends from throat to proper radial distance +/- ∞ Density, tension and pressure vanish asymptotically Material is everywhere exotic i.e., 0 everywhere Violates energy conditions Need quantum field theory “Catenoid” Exotic Matter Limited to Throat 2 (r b0 ) b b0 1 ; 0 for b0 r b0 a0 a0 b 0 for r b0 a0 Spacetime flat at r > b0 + a0 Tidal forces bearable Absurdly Benign Wormhole Travel time reasonable BUT throat radius must be large to have meaningful wormhole Backward Time Travel From H.G.
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