Einstein Toolkit Workshop Joshua Faber Apr 05 2012
Outline Space, time, and special relativity The metric tensor and geometry Curvature Geodesics Einstein s equations The Stress-energy tensor 3+1 formalisms Schwarzschild black holes and coordinates Linearized gravity and gravitational radiation
Other sources For a very nice treatment of GR available freely online, I suggest Sean Carroll s notes: gr-qc/9712019. For the best scientific biography of Einstein, including a discussion of the divergenceless of various tensors with full equations, I suggest Subtle is the Lord, by Abraham Pais.
Space and time in Newtonian physics In Newtonian physics, there is space, and there is time, and they are different. Gravity is instantaneous (note that t does not appear): a g = Φ; Φ( r) = Gρ( r ) r r d 3 r (1) Galilean relativity: The laws of physics are unchanged for observers in different inertial frames, i.e., observers moving at constant velocities.
Maxwell s equations Gravity and light were thought to need some form of medium through which they travel: It is inconceivable, that inanimate brute matter should, without the mediation of anything else, which is not material, operate upon and affect other matter without mutual contact. Newton Maxwell s equations seemingly violate Galilean invariance; they predict waves moving at the speed of light, but with respect to what? Seemingly, the ether, whatever it was.
Special relativity Famously, it was derived not from the Michelson-Morley experiment, but as a way to understand electromagnetism. Joshua Faber Apr 05 2012
Special relativity Assume the speed of light is constant, and consider a beam of light to a moving observer: Fitzgerald-Lorentz length contraction, Lorentz transformations, etc. - Note that the pieces were being assembled by many, but Einstein provided the overarching framework
Special relativity: Lorentz transformations According to special relativity, moving observers see time differently (dilation) and space differently in the direction of motion (length contraction): t = γ(t βx); x = γ(x βt) (β v/c) (2) Note that two observers moving at different speeds mutually agree that the other one s clock ticks too slowly, and has a shorter ruler in the direction of motion. Lorentz boosts can be written in matrix form: Λ µ ν = where φ = tanh 1 β. cosh φ sinh φ 0 0 sinh φ cosh φ 0 0 0 0 1 0 0 0 0 1 ; (x ) µ = [ 3 ] Λ µ νx ν (3) ν=0
Special relativity: Invariants All observers in SR agree on the speed of light, but also the invariant distance: ds 2 = η µν dx µ dx n u = dt 2 + dx 2 + dy 2 + dz 2 (4) where we define the Minkowski metric η µν = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 (5) Lorentz transformations leave this quantity invariant. Points with ds 2 > 0 are called spacelike, and are causally disconnected; ds 2 < 0 is timelike, and causally connected; ds 2 = 0 is called lightlike, and describes possible photon paths.
Special relativity: spacetime diagrams
General relativity: the equivalence principle As long as we restrict ourselves to purely mechanical processes in the realm where Newton s mechanics holds sway, we are certain of the equivalence of the systems K [uniformly accelerating] and K [homogeneous gravitational field]. But this view of ours will not have any deeper significance unless the systems K and K are equivalent with respect to all physical processes, that is, unless the laws of nature with respect to K are in entire agreement with those with respect to K. By assuming this to be so, we arrive at a principle which, if it is really true, has great heuristic importance. For by theoretical consideration of processes which take place relatively to a system of reference with uniform acceleration, we obtain information as to the career of processes in a homogeneous gravitational field.
General relativity: the equivalence principle Gravity curves paths, toward the gravitating mass(-energy). Orbiting bodies think they are traveling in straight lines.
General relativity: the metric tensor The metric tensor determines the geometry of spacetime: ds 2 = g µν dx µ dx ν (6) where the metric tensor g µν must be symmetric, g µν = g νµ (10 free components). What is a tensor, though?
Vectors In Riemannian geometry, vectors are defined via arguments about tangent spaces: ( V )f = V µ f = some value that depends on f (7) x µ If we change coords to primed coords, the value of this operator, which is just a number, has to remain invariant: and we conclude that V ν f = V x ν ν x µ x ν x µ f (8) V µ ν x µ = V x ν The transformation properties of an object when we change the coords define a vector. Note that this definition works well for displacements: dx µ = x µ dx ν (9) x ν
Covariant Vectors and Tensors There are a number of objects that operate linearly on vectors. These include things like gradients themselves: µ ν ν x x = x µ x µ = x ν x µ ν (10) Tensors generalize these operations. We transform each index separately when we change coordinates. Consider the metric tensor, which is a linear operator acting on two vectors, like a dot product: ( ) ( ) x ds 2 = g µν dx µ dx ν µ x ν = g µν dx α dx β = g α x α x β β dx α dx β so we find g α β = g x µ µν x α x ν x β g µν = g α β x α x µ x β x ν (11)
Christoffel symbols Partial derivatives of tensors are not tensors: W ν x µ = x µ ( ) x ν x µ x µ W ν x ν = x µ x µ x ν x ν ( x µ W ν ) + W ν x µ x µ x ν x µ x ν where the first term looks right, but the second term, containing derivatives of changes in coordinates, do not work. Instead we can introduce the covariant derivative, which eliminates the term that represents variations in a coordinate, rather than a vector field µ W ν = W ν x µ Γ α µ ν W α (12) where the Christoffel symbols capture the coordinate derivatives as follows Γ α βγ = 1 2 g ασ ( β g γσ + γ g βσ σ g βγ ) (13) This can be generalized for tensors of all kinds.
Curvature If you move around in flat space, keeping the covariant derivative of a vector zero, nothing happens. In curved space, thing do! dv a = R a bcd v b dp c dq d (14)
Geodesics But wait, what is a straight line in curved space, anyway? To move any vector V µ along a path x µ (t), we introduce the notion of parallel transport: x V = 0 d dt v µ + Γ µ dx α αβ dt v β = 0 (15) An unaccelerated path is a curve that parallel transports itself with speed ẋ ν = V µ : d 2 x µ dt 2 = d dt v µ + Γ µ αβ v α v β (16) Parallel transport leaves the length of a vector invariant, d dt [g µνv µ v ν ] = 0.
Gravity and curvature Einstein s idea for the general theory of relativity is that mass-energy curves space. Mass-energy is characterized by the stress-energy tensor: ρ 0 0 0 T µν = (ρ 0 [1 + ɛ] + p)u µ u ν + pη µν = 0 p 0 0 0 0 p 0 (17) 0 0 0 p where ρ 0 is the rest-mass density, ρ = ρ 0 [1 + ɛ] is the mass-energy density, ɛ the internal energy density of a fluid, p the pressure, u µ the 4-velocity. We are assuming an ideal fluid, measured in its own rest frame, for the matrix expression. The stress energy tensor encodes not only the density of mass and momentum, but also their flows (the pressure term). The evolution set is: µ (ρu µ ) = 0; µ T µν = 0 (18)
Einstein s equations Einstein s equations demonstrate how mass curves space: R µν 1 2 g µνr = 8πGT µν (19) Put simply, the left hand side involves second derivatives, both in time (accelerations) and space (just like Gauss s law for gravitation). The right hand-side incorporates the gravitational contributions of mass, and momentum, and stresses in general. Because we have a second-order system, we need both the metric and its first derivatives to be given as initial data to evolve the system. The left hand side is automatically divergence free, and this fact played a critical role in deriving it in the first place: µ (R µν 1 2 g µνr) = 0 (20)
3+1 formalism Since we generally think of relativistic systems as encompassing space and evolving forward in time, it is often useful to perform a 3+1 split, where a time coordinate is identified. The general form of the metric, from which we can determine the Christoffel symbols, curvature tensor, etc. is ds 2 = (α 2 β i β i )dt 2 + 2β i dx i dt + γ ij dx i dx j (21) The 3-metric γ ij, with 6 independent components, determines the spatial geometry at a single moment in time, known as a slice. The lapse function α and shift vector β i, are known as gauge quantities, and represent a freely-specifiable choice as to how coordinates evolve.
3+1 formalism Much of numerical relativity comes from determining how to do the following steps: The 3+1 formalism is not numerically stable, but by evolving additional variables whose behavior we should know, it can stabilize the system: t γ ij =... (22) We need an evolution for the time derivative of the (spatial) metric. This is captured by the extrinsic curvature K ij, which is closely related to the metric derivatives. Think of it as how the metric evolves from slice to slice. We need gauge evolution equations for α and β i. We can choose these freely, but our choice affects the resulting evolution set. This is the choice of coordinates for the spacetime.
Black holes The most familiar black hole solution, for an uncharged BH without any angular momentum, is the Schwarzschild solution: ( ds 2 = 1 2GM ) ( dt 2 + 1 2GM ) 1 dr 2 + r 2 dω 2 (23) r r where dω 2 = dθ 2 + sin 2 θdφ 2. This is a stationary solution to Einstein s equations. At r = 2M, we have a singularity, since there is no change in distance in time, and the radial term goes to infinity. This is the BH horizon. While indeed light cannot escape from the horizon, the singularity is an artifact of the coordinates we chose. If we let R = 1 2 (r M + r(r 2M)), we find the metric takes the isotropic form: ds 2 = ( 1 M/2R 1 + M/2R ) 2 dt 2 + ( 1 + M 2R ) 4 [dr 2 + r 2 dω 2 ] (24) and we have removed the radial singularity (note that r = 2M R = M/2).
Linearized gravity In most places not near black holes or neutron stars, curvature is not so extreme. In these case, we can assume linearized gravity: g µν = η µν + h µν ; h µν 1 (25) In the proper gauge, we can reduce Einstein s equations to the form: h µν = ( 2 t + 2 )h = 16πGT µν (26) This is the wave equation, resulting in the prediction of gravitational waves. It can be shown they are transverse, like EM waves, but with two polarizations: