Gravitation: Special Relativity

Transcription

1 An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013

2 Space and Time Space-time: is a 4-dim set, with elements labeled by 3-dim of space and one of time. Event: An individual point in space-time. Worldline: A path of events. There is a difference between the paths that particles follow in Special Relativity (SR) and those in Newton s theory. The absence of a preferred time-slicing foliation is fundamental in SR. t x, y, z space at a fixed time

3 y y! s! y! y x! x! x x Geometrical facts about the spatial planes are independent of our choice of coordinates. E.g., the distance between two points: ( s) 2 = ( x) 2 + ( y) 2. In a different Cartesian coordinate system x and y the distance is unaltered: ( s) 2 = ( x ) 2 + ( y ) 2. S is invariant under such changes of coordinates.

4 Space-time interval ( s) 2 = c 2 ( t) 2 + ( x) 2 + ( y) 2 + ( z) 2. Notice: s can be positive, negative, or zero. c is some fixed conversion factor between space and time. c is the conversion factor that makes s invariant. The minus sign is necessary to preserve invariance. In a more compact form, using the summation convention ( indices which appear both as superscripts and subscripts are summed over), form. ( s) 2 = η µν x µ x ν. where x µ = ( t, x, y, z) and η µν is a 4 4 matrix called the metric: c η µν =

5 ( s) 2 = η µν x µ x ν Light Cone: Time-like separated if ( s) 2 < 0 Space-like separated if ( s) 2 > 0 Null or Light-like separated if( s) 2 = 0 Proper Time τ: Measures the time elapsed as seen by an observer moving on a straight path between events. That is c 2 ( τ) 2 = ( s) 2 Notice: if x i = 0 then c 2 ( τ) 2 = η µν ( t) 2, thus τ = t.

6 What kind of transformations leave s invariant? Translations: x µ x µ = x µ + a µ, where a µ four fixed numbers. Rotations : or x µ = Λ µ ν x ν, x = Λx. ( s) 2 = ( x) T η( x) = ( x ) T η ( x ) = ( x) T Λ T η Λ( x), and therefore or η = Λ T η Λ, η ρσ = Λ µ ρλ ν ση µ ν. Goal: To find Λ µ ν such that the components of η µ ν are the same as those of η ρσ.

7 Lorentz Transformations: the matrices Λ µ ν that satisfy η = Λ T η Λ Lorentz transformations categories: Rotations: Λ µ ν = 0 cos θ sin θ 0 0 sin θ cos θ Boosts or rotations between space and time directions. cosh φ sinh φ 0 0 Λ µ ν = sinh φ cosh φ The set of both translations and Lorentz transformations is a 10-parameter non-abelian group (4+3+3), called the Poincaré group.

8 For the boost transformation above, t = t cosh φ x sinh φ x = t sinh φ + x cosh φ. Consider the point x = 0. Is is moving at v = x t = sinh φ cosh φ = tanh φ. Defining v = tanh φ and γ = 1/ 1 v 2 one obtains the conventional expressions for Lorentz transformations: t = γ(t vx) x = γ(x vt) Notice: We have set c = 1

9 Inertial Frames Inertial Frame: a frame in which Newton s first law holds; that is, Inertial Coordinates: d 2 x i dt 2 = 0 Build a rigid frame and label the grid points with coordinates (x, y, z); thus at each point d 2 x i dt 2 = 0 Place a clock at each grid point and synchronize them

10 Clock Synchronization Send light from your location to a neighboring point. Reflect the light from the neighboring point back to your location. Define the time at the neighboring point as τ 2 = τ 3 + τ 1 2 In this way, a single observer s clock can be used to define temporal coordinates everywhere.

11 In Newtonian spacetimes, F 1 and F 2 are related by Galilean transformations t = t x = x vt y = y z = z In SR spacetimes, F 1 and F 2 are related by Lorentz transformations F 1 and F 2 are inertial frames S are hypersurfaces of simultaneity t = γ(t vx) x = γ(x vt) y = y z = z Inertial frames can only differ by a rotation, translation and uniform motion

12 O 1 and O 2 are at rest O 3 is in uniform motion O 4 is accelerating O 5 and O 6 orbiting about P, which is at rest O 7 and O 8 orbiting about P, which is in uniform motion O 1 and O 2 are in uniform motion O 3 is at rest O 4 is accelerating O 5 and O 6 orbiting about P, which is in uniform motion O 7 and O 8 orbiting about P, which is at rest Principle of Relativity Identical experiments carried out in different inertial frames give identical results.

13 Vectors A vector is located at a given point in space-time p T p manifold M Tangent space T p at p is the set of all possible vectors located at that point. A vector space is a collection of vectors-objects such that for any two vectors V and W and real numbers a and b, (a + b)(v + W ) = av + bv + aw + bw. Vector field is set of vectors with exactly one at each point in spacetime. Tangent bundle T (M) is set of all the tangent spaces of a manifold M. A basis is any set of vectors which both spans the vector space and is linearly independent.

14 Consider at each T p a basis ê (µ) adapted to the coordinates x µ ; that is, ê (1) pointing along the x-axis, etc. Then, any abstract vector A can be written as A = A µ ê (µ). The coefficients A µ are the components of the vector A. The real vector is the abstract geometrical entity A, while the components A µ are just the coefficients of the basis vectors in some convenient basis. The parentheses around the indices on the basis vectors ê (µ) label collection of vectors, not components of a single vector.

15 Tangent to a vector curve Consider a parametrized curve or path x µ (λ). Its tangent vector V (λ) has components Thus V = V µ ê (µ). Under a Lorentz transformation V µ = dxµ dλ. x µ x µ = Λ µ ν x ν therefore V µ V µ = Λ µ ν V ν. However, the vector itself is invariant under Lorentz transformations. That is, V = V µ ê (µ) = V ν ê (ν ) = Λ ν µv µ ê (ν ). Therefore, ê (µ) = Λ ν µê(ν ) Let Λ ν µ denote the inverse to Λ ν µ; that is, such that where δ ρ µ is the Kronecker delta. Thus, Λ ν µ Λ σ µ = δ σ ν, Λ ν µ Λ ν ρ = δ µ ρ, ê (ν ) = Λ ν µ ê (µ).

16 Dual Vectors or 1-forms Co-tangent space Tp : dual space to Tp. Dual vector space: is the space of all linear maps from the original vector space to the real numbers. That is, if ω Tp is a dual vector, then ω(av + bw ) = aω(v ) + bω(w ) R, where V, W are vectors and a, b are real numbers. The dual vectors or one-forms yield also a vector space. That is if ω and η are dual vectors, we have (a + b)(ω + η) = aω + bω + aη + bη.

17 Dual Vectors or 1-forms There exists a set of basis dual vectors ˆθ (ν) by demanding ˆθ (ν) (ê (µ) ) = δ ν µ. Therefore ω = ω µ ˆθ(µ). Elements of T p are also referred as contravariant vectors and elements of T p The action of a one-form on a vector: as covariant vectors. ω(v ) = ω µv ν ˆθ (µ) (ê (ν) ) = ω µv ν δ µ ν = ωµv µ R. Vectors are also linear maps on dual vectors, i.e. V (ω) ω(v ) = ω µv µ. Therefore, the dual space to the dual vector space is the original vector space itself.

18 Transformation properties of dual vectors: ω µ = Λ µ ν ω ν, and for basis dual vectors, ˆθ (ρ ) = Λ ρ σ ˆθ (σ). Simplest example of a dual vector is the gradient of a scalar function: dφ = φ x µ ˆθ (µ). with a transformation rule µ φ x φ = x µ x µ x µ = Λ µ µ φ x µ, Shorthand notations for partial derivatives: φ = µφ = φ, x µ µ. Notice also µφ xµ λ = dφ dλ.

19 Tensors A tensor T of type (or rank) (k, l) is a multilinear map from a collection of dual vectors and vectors to R: T : Tp T p Tp Tp R (k times) (l times) denotes the Cartesian product; e.g., T p T p is the space of ordered pairs of vectors. Multilinearity means that the tensor acts linearly in each of its arguments; e.g., for a tensor of type (1, 1), T (aω + bη, cv + dw ) = act (ω, V ) + adt (ω, W ) + bct (η, V ) + bdt (η, W ). A scalar is a type (0, 0) tensor, a vector is a type (1, 0) tensor, and a dual vector is a type (0, 1) tensor. The space of all tensors of a fixed type (k, l) forms a vector space; that is, they can be added together and multiplied by real numbers.

20 Tensor Product The tensor product of a (k, l) tensor T with a (m, n) tensor S, denoted by T S, is defined by (T S)(ω (1),..., ω (k),..., ω (k+m), V (1),..., V (l),..., V (l+n) ) = T (ω (1),..., ω (k), V (1),..., V (l) )S(ω (k+1),..., ω (k+m), V (l+1),..., V (l+n) ). Steps: act T on the appropriate set of dual vectors and vectors act S on the remainder dual vectors and vectors multiply the answers. Notice, in general, T S S T. Examples Given two dual-vectors U and V (U V )(ê (µ), ê (ν) ) = U(ê (µ) )V (ê (ν) ) = U α ˆθ(α) (ê(µ) )V β ˆθ (β) (ê (ν) ) = U αδ α µ V β δ β ν = UµVν Given a tensor T of rank (1,1) and a tensor S of rank (2,1), their tensor product is (T S)( ˆθ (µ), ê (ν), ˆθ (α), ˆθ (β), ê (γ) ) = T µ ν S αβ γ

21 Tensor Basis Recall for a vector T Also T ( ˆθ (µ) ) = T α ê (α) ( ˆθ (µ) ) = T α δ µ α = T µ (ê (α) ê (β) )( ˆθ (µ), ˆθ (ν) ) = ê (α) ( ˆθ (µ) )ê (β) ( ˆθ (ν) ) = δ µ α δν β We can then view (ê (α) ê (β) ) as the basis for rank (2,0) tensors. Thats is, T = T αβ (ê (α) ê (β) ) The components of T can be found from T ( ˆθ (µ), ˆθ (ν) )) = T αβ (ê (α) ê (β) )( ˆθ (µ), ˆθ (ν) )) = T αβ δ µ α δν β = T µν In general, the basis for the space of all (k, l) tensors is constructed by taking tensor products of basis vectors and dual vectors; namely An arbitrary (k, l) tensor is then written as ê (µ1 ) ê (µ k ) ˆθ (ν 1 ) ˆθ (ν l ). T = T µ 1 µ k ν1 ν l ê (µ1 ) ê (µ k ) ˆθ (ν 1 ) ˆθ (ν l ). and its components as T µ 1 µ k ν1 ν l = T ( ˆθ (µ 1 ),..., ˆθ (µ k ), ê (ν1 ),..., ê (ν l ) ).

22 More Tensor Properties A (k, l) tensor has k upper indices and l lower indices; that is, T µ 1 µ k ν1 ν l The order of the indices is important; that is, T µνρ σν T µρν σν Lorentz transformations: T µ 1 µ k ν 1 ν l = Λ µ 1 µ1 Λ µ k µk Λ ν 1 ν 1 Λν l ν l T µ 1 µ k ν1 ν l Tensor Contractions T µ ν : V ν U µ = T µ ν V ν T µρ σ : S σ ρν U µ ν = T µρ σs σ ρν

23 More Tensor Properties Inner or dot product: Given the metric η, η(v, W ) = η µν V µ W ν = V W If η(v, W ) = 0, the vectors are orthogonal. Since η(v, W ) = V W is a scalar, it is left invariant under Lorentz transformations. norm of a vector is given by V V. if η µν V µ V ν < 0, V µ is timelike is = 0, V µ is lightlike or null > 0, V µ is spacelike.

24 Important Tensors Kronecker delta δ ν µ = diag(1, 1, 1, 1), of type (1, 1), Inverse metric η µν, a type (2, 0) such that η µν η νρ = η ρν η νµ = δ ρ µ Levi-Civita tensor, a (0, 4) tensor: +1 if µνρσ is an even permutation of 0123 ɛ µνρσ = 1 if µνρσ is an odd permutation of otherwise. Electromagnetic field strength tensor, a (0, 2) tensor: 0 E 1 E 2 E 3 F µν = E 1 0 B 3 B 2 E 2 B 3 0 B 1 = Fνµ E 3 B 2 B 1 0

25 Manipulating Tensors S µρ σ = T µνρ σν T αβµ δ = η µγ T αβ γδ T µ β γδ = η µαt αβ γδ T µν ρσ = η µαη νβ η ργ η σδ T αβ γδ V µ = η µν V ν ω µ = η µν ω ν

26 Maxwell s Equations recall F 0i = E i B t E = 4πJ E = 4πρ E + t B = 0 B = 0 in component notation F ij = ɛ ijk B k then, the first two Maxwell eqs. read j F ij 0 F 0i = 4πJ i i F 0i = 4πJ 0. ɛ ijk j B k 0 E i = 4πJ i i E i = 4πJ 0 or equivalently µf νµ = 4πJ ν ɛ ijk j E k + 0 B i = 0 i B i = 0. similarly the last two Maxwell eqs. can be rewritten as [µ F νλ] = 0

27 Energy and Momentum Four-velocity U µ = dxµ dτ. Since dτ 2 = η µν dx µ dx ν, the four-velocity is time-like, i.e., η µν U µ U ν = 1 In the rest frame of a particle, its four-velocity has components U µ = (1, 0, 0, 0). Four-momentum: p µ = mu µ with m the rest-mass of the particle. Energy of the particle is given by E = p o = m. Recall E = m c 2. Notice that p µ = mu µ = m dxµ where γ = dt/dτ and V µ = (1, dx i /dt) = (1, v) dτ = m dt dx µ = mγv µ dτ dt Notice also, from 1 = η µν U µ U ν = γ 2 η µν V µ V ν = γ 2 ( 1 + v 2 ) thus γ = 1/ 1 v 2 the Lorentz boost factor.

28 Newton s 2nd Lay and Special Relativity Newton s Second Law: f = m a = d p dt Lorentz Law: f = q( E + v B) in SR f µ = m d2 dτ 2 xµ (τ) = d dτ pµ (τ) in SR f µ = qu λ F λ µ.