A Brief Introduction to Mathematical Relativity Arick Shao Imperial College London Arick Shao (Imperial College London) Mathematical Relativity 1 / 31
Special Relativity Postulates and Definitions Einstein’s Postulates (A. Einstein, 1905) Postulates of special relativity: ∗ Relativity principle: The laws of physics are the same in 1 all inertial frames of reference. Speed of light: The speed of light in vacuum has the 2 same value c in all inertial frames of reference. Postulates + physical considerations ⇒ : Observers moving at different velocities will perceive length, time, etc., differently. A. Einstein (1879–1955) ⋆ ∗ Quoted from Nobelprize.org. ⋆ Photo from Nobelprize.org. Arick Shao (Imperial College London) Mathematical Relativity 2 / 31
Special Relativity Postulates and Definitions Minkowski’s Formulation (1907) Hermann Minkowski: Geometric formulation of special relativity. Ideas later extended to general relativity. Time ( R ) + space ( R 3 ) = spacetime ( R 4 ) More accurately, R 4 “modulo coordinate systems.” Formally, R 4 as a (differential) manifold . (Newtonian theory: R × R 3 ) H. Minkowski (1864–1909) ⋆ ⋆ Photo from www.spacetimesociety.org . Arick Shao (Imperial College London) Mathematical Relativity 3 / 31
Special Relativity Postulates and Definitions Euclidean vs. Minkowski 4-d Euclidean space ( R 4 , δ ) : 4-d Minkowski spacetime ( R 4 , η ) : Euclidean (square) distance: Minkowski (square) “distance”: 4 3 � � ( p k − q k ) 2 . d 2 ( p , q ) := d 2 ( p , q ) := −( q 0 − p 0 ) 2 + ( q k − p k ) 2 . k = 1 k = 1 Corresponding differential structure Corresponding differential structure ( Euclidean metric ): ( Minkowski metric ): δ := dx 2 + dy 2 + dz 2 + dw 2 . η := − dt 2 + dx 2 + dy 2 + dz 2 . For vectors u , v ∈ R 4 : For vectors u , v ∈ R 4 : 4 3 � � η ( u , v ) = − u 0 v 0 + u k v k . u k v k . δ ( u , v ) = k = 1 k = 1 Riemannian manifold Lorentzian manifold Arick Shao (Imperial College London) Mathematical Relativity 4 / 31
Special Relativity Postulates and Definitions Causal Character Geometry of ( R 4 , η ) radically different from that of ( R 4 , δ ) . Lack of sign definiteness ⇒ different directions have different meanings. Causal character : A vector v ∈ R 4 is Spacelike if η ( v , v ) > 0 or v = 0. Timelike if η ( v , v ) < 0. Null ( lightlike ) if η ( v , v ) = 0 and v � = 0. Physical interpretations: Observer : timelike curve. Light : null lines. ∗ Image by Stib on en.wikipedia.org . Arick Shao (Imperial College London) Mathematical Relativity 5 / 31
Special Relativity Consequences Relativity Many concepts have no absolute prescription: Elapsed time, length, energy-momentum. Only makes sense relative to an observer. Observer ¯ z ) adapted to ¯ O ⇒ coordinates ( ¯ t , ¯ x , ¯ y , ¯ O . z = 0 along ¯ x = ¯ y = ¯ O . ¯ Observer can measure with respect to these coordinates . Constant velocity ⇒ inertial coordinate system : t 2 + d ¯ x 2 + d ¯ y 2 + d ¯ z 2 . η = − d ¯ Arick Shao (Imperial College London) Mathematical Relativity 6 / 31
Special Relativity Consequences Simultaneity Observers moving at different (constant) velocities will perceive different events to be “at the same time.” A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 ¯ t = ¯ c A 0 B 0 A 0 t = c ¯ t = ¯ c B 0 t = c Coordinates with observer A at rest. Coordinates with observer B at rest. Arick Shao (Imperial College London) Mathematical Relativity 7 / 31
Special Relativity Consequences Length Contraction Observers moving at different velocities perceive lengths differently. Shaded region represents rod. A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 t = ¯ ¯ c A measures “length” of blue bolded A 0 segment through rod. t = c B 0 B measures “length” of red bolded segment through rod. B measures shorter length than A. Observers A and B measure a rod (at rest with respect to A ). Arick Shao (Imperial College London) Mathematical Relativity 8 / 31
Special Relativity Consequences Time Dilation Clocks moving at different velocities observed to tick at different speeds. Both A and B carry clock. A : x = y = z = 0 B : ¯ x = ¯ y = ¯ z = 0 Both clocks synchronised at O . A 0 A measures both clocks at t = c . t = c B 0 A measures less time elapsed on B’s O clock than A’s clock. Observer A measures clocks carried by both A and B . Arick Shao (Imperial College London) Mathematical Relativity 9 / 31
Special Relativity Consequences Twin Paradox Different timelike curves between two events will have different lengths. t ¯ B t η =− dt 2 + dx 2 + dy 2 + dz 2 t 2 + d ¯ x 2 + d ¯ y 2 + d ¯ z 2 η � =− d ¯ A From A to B : more time elapses for t -observer than for ¯ t -observer. Arick Shao (Imperial College London) Mathematical Relativity 10 / 31
General Relativity Postulates and Definitions Geometry and Gravity Special relativity does not (A. Einstein, 1915) General relativity : model gravity. Gravity not modeled as a force, but rather through geometry of spacetime. Revolutionary idea: gravity ⇔ curvature Curved spacetime, with gravity represented by spacetime curvature. ∗ ∗ Image by Johnstone on en.wikipedia.org . A. Einstein (1879–1955) Arick Shao (Imperial College London) Mathematical Relativity 11 / 31
General Relativity Postulates and Definitions Spacetimes Extend notion of spacetime: ( R 4 , η ) � → 4-dimensional Lorentzian manifold ( M , g ) . Geometric content: Lorentzian metric g on M . g has “same signature (− 1 , 1 , 1 , 1 ) ” as η . ∗ Study of spacetimes ⇔ Lorentzian geometry : Analogue of Riemannian geometry. Lines in R 4 � → geodesics Can formally make sense of curvature . ∗ At each p ∈ M , we have a bilinear form g | p on T p M of signature (− 1 , 1 , 1 , 1 ) . Arick Shao (Imperial College London) Mathematical Relativity 12 / 31
General Relativity Postulates and Definitions Physical Interpretations Principle of covariance : physical laws are intrinsic properties of the manifold ( M , g ) , i.e., independent of coordinates on M. Causal character for tangent vectors: Observers : timelike curves. v is spacelike if g ( v , v ) > 0 or v = 0. Free fall: timelike geodesics. v is timelike if g ( v , v ) < 0. Light: null geodesics. v is null if g ( v , v ) = 0 and v � = 0. Arick Shao (Imperial College London) Mathematical Relativity 13 / 31
General Relativity Postulates and Definitions Matter Fields Gravity closely coupled to matter via the Einstein field equations : Ric g − 1 2 Sc g g = T . Ric g : Ricci curvature associated with g . Sc g : Scalar curvature associated with g . T : Stress-energy tensor associated with matter field Φ . Φ : satisfies equations according to its physical theory. No matter field ⇒ Einstein-vacuum equations ( EVE ): Ric g = 0. Arick Shao (Imperial College London) Mathematical Relativity 14 / 31
General Relativity The Einstein-Vacuum Equations Connection to Differential Equations Question How do we interpret the EVE? Write equations in terms of g and a fixed coordinate system on M : 2nd-order quasilinear system of PDE for components of g : � 0 = − 1 g αβ ( ∂ α ∂ β g µν − ∂ β ∂ ν g µα − ∂ β ∂ µ g να + ∂ µ ∂ ν g αβ ) (1) 2 α,β � + 1 g µν g αβ g γδ ( ∂ α ∂ β g γδ − ∂ β ∂ γ g αδ ) + F 0 ( g , ∂ g ) . 2 α,β,γ,δ Q. What is the character of (1)? (elliptic, parabolic, hyperbolic) Determines what types of problems are reasonable to solve. Arick Shao (Imperial College London) Mathematical Relativity 15 / 31
General Relativity The Einstein-Vacuum Equations Special Coordinates Bad news: In general, (1) is none of the above. In special coordinates, (1) becomes hyperbolic. � 0 = − 1 g αβ ∂ α ∂ β g µν + F 1 ( g , ∂ g ) . (2) 2 α,β Y. Choquet-Bruhat (b. 1923) ⋆ Should be solved as an “initial value problem”. (1952, Y. Choquet-Bruhat) Solved Einstein-vacuum ⋆ Photo by Renate Schmid for the equations for short times. Oberwolfach Photo Collection ( owpdb.mfo.de ). Arick Shao (Imperial College London) Mathematical Relativity 16 / 31
General Relativity The Einstein-Vacuum Equations Well-Posedness Question Is the initial value problem well-posed ? Given initial data, can we: Show existence of solution to EVE? 1 Show uniqueness of this solution? 2 Show continuous dependence of solution on initial data? 3 In other words, given the state of the universe at some time, can we: (1) + (2): Predict the future/past? (3): Approximately predict the future/past? Arick Shao (Imperial College London) Mathematical Relativity 17 / 31
General Relativity The Einstein-Vacuum Equations Solving the Equations Many difficulties behind solving the EVE: Equations are highly nonlinear. Initial data must first satisfy (elliptic) constraint equations. Note: Unlike other PDE, we are solving for the spacetime itself ! Usually, solve for functions on fixed background (e.g., R N ). Here, we solve for ( M , g ) , i.e., the “universe”. Example Initial data: Euclidean space ( R 3 , δ ) Solution: Minkowski spacetime ( R 4 , η ) Arick Shao (Imperial College London) Mathematical Relativity 18 / 31
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