Curved Spacetime I. Dr. Naylor

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2 Curved Spacetime I Dr. Naylor

3 Last Week Einstein's principle of equivalence We discussed how in the frame of reference of a freely falling object we can construct a locally inertial frame (LIF) Space tells matter how to move Matter tells space how to curve John Wheeler To find a way to connect different LIFs we need the concept of a Manifold A manifold is a continuous space whose points can be assigned coordinates The number of dimensions being the number of coords For example S 2 and S 4 A manifold is differentiable if we can define a scalar field φ at each point which can be differentiated everywhere 2

4 Riemannian Manifolds A Riemannian manifold is a differentiable manifold with a symmetric metric tensor (g αβ = g βα ) at every point such that for example in Euclidean S 3 However in special and general relativity is of indefinite sign Remember in SR the metric can be spacelike, timelike or null, i.e., ds 2 >0, <0, =0 In this case is called a Pseudo-Riemannian metric For general coordinates {x α }, the interval between two points is ds 2 = g αβ dx α dx β, where in SR we have g αβ = η αβ 3

5 LIFs & the metric In GR we cannot use η αβ Theorem: any symmetric matrix (at any given point x 0 ) can be converted into a diagonal form which is locally Minkowski: In general Λ α β will not diagonalize g αβ at every point because there are 10 functions for only four transformation functions X α (x β ) Note the following theorem: We can always choose Λ α β such that g αβ,γ = 0 is satisfied ) g αβ (x µ )=η αβ +O [(x µ ) 2 ] Thus, any LIF is a frame where g αβ (x 0 )=η αβ ; g αβ,γ (x 0 )= 0 and g αβ,γµ (x 0 ) 0 4

6 Christoffel symbols In Minkowski spacetime the derivative of a vector Because the basis vectors do not vary; however in a general curved spacetime vary from point to point ) But since is a vector for a given β this can be written as a linear combination of the bases: These Γ are known as Christoffel symbols (or the metric connection) and hence 5

7 Thus, we write Covariant derivatives the semicolon ; denotes the covariant derivative The components of V α ;β are a tensor defined by Under a coordinate transformation Λ α β = x α,β we have hence the name covariant derivative. Note the normal derivative, is not a tensor unless in Minkowski space: Only zero for Minkowski 6

8 Reasoning? Thus, we add Γ α βγ to the derivative of a vector so that it remains a tensor in any spacetime. Namely, by defining Then the ; derivative transforms with the bases and hence is called the covariant derivative: Note that the Christoffel symbol Γ is also known as the metric connection This point will be explained later 7

9 Covariant derivatives on other tensors For scalars we have because scalars do not depend on basis vectors For a one form (covector) we have and therefore implying the components transform p α;β = p α,β - p µ Γ µ αβ For general tensors we then have 8

10 Γ α βγ from the metric Consider V α ;β We can lower the upper index using the metric g αβ : V α;β = g αµ V µ ;β but by linearity we must have For consistency this implies Finally assuming that Γ γ αβ = Γ γ βα then we have Exercise: by writing different permutations of the above equation and assuming g αγ g αν = δ γ ν show that 9

11 Parallel Transport Definition: A vector field is parallel transported along a curve with tangent in an inertial frame if and only if λ parameterizes the curve and is usually taken to be the proper time τ Thus in a general coordinate frame we have We use the rule comma goes to semicolon : That is work things out in a LIF, and if it is a tensor equation then generalize to ; e Remember by using the Christoffel symbol we found a way to define in a general frame Thus, from an LIF we can use,! ; 10

12 Geodesic equation Definition: A curve λ is a geodesic if it parallel transports its own tangent vector: V U Parallel transport Geodesic A geodesic is the closest thing to a straight line in a curved space. Since U α =dx α /dλ and d/dλ=u β / x β then we obtain the geodesic equation This is a 2 nd order differential equation for x α (λ) so is unique as long as the initial position x 0 and velocity U 0 are specified U 11

13 Relation to the equivalence principle (EP)? In Minkowski space d 2 x α / dτ 2 =0; however, even in flat spacetime if we use a non-inertial frame, for example polar coordinates, then in this case Γ equates to inertial forces due to a flat metric, but with g αβ η αβ. By the EP we know that gravitational forces and inertial forces are related (see last week s lecture) Furthermore gravity requires a curved spacetime Thus, we use the metric connection Γ for a non-flat metric g αβ where the Γ s are interpreted as force terms Implying that g αβ behaves like the Newtonian potential Φ Φ satisfies a 2 nd O.D.E. and analogously so does g αβ 12

14 Next week s class: Curvature Thus we can obtain a relativistic theory of gravitation once we find this set of partial differential equations Note that unlike with the Newtonian differential equation we are dealing with tensors which are a set of partial differential equations Before doing this though we must learn how to quantify the amount of curvature... Next week 13