General Relativity and Differential

Transcription

1 Lecture Series on... General Relativity and Differential Geometry CHAD A. MIDDLETON Department of Physics Rhodes College November 1, 2005 OUTLINE Distance in 3D Euclidean Space Distance in 4D Minkowski Spacetime Principle of Equivalence Distance in 4D Non-Euclidean Spacetime (metric tensor, Christoffel symbols, Einstein Field Equations)

2 The General Theory of Relativity 1 The General Theory of Relativity General Relativity is the most beautiful physical theory ever invented. - Spacetime and Geometry One of the Greatest Achievements of the Human Mind. - Introducing Einstein s Relativity

3 3D Euclidean Space 2 Line element (distance) in Euclidean space lim 0 ( s)2 = ( x) 2 + ( y) 2 + ( z) 2 ds 2 3D = dx2 + dy 2 + dz 2 = d x d x z (1) ds dz dx y x dy ds 2 3D is the line-element measuring distance Pythagorean Theorem ds 2 3D is invariant under rotations

4 3D Euclidean Space 3 Line element using matrix notation ds 2 3D = δ ijdx i dx j where δ ij = (2) Matrix multiplication Repeated index sum over index ds 2 3D = δ ijdx i dx j = 3 3 i=1 j=1 δ ij dx i dx j x i,j : x y z

5 3D Euclidean Space 4 ds 2 3D = δ ijdx i dx j Summing over the i index... ds 2 3D = δ 1jdx 1 dx j + δ 2j dx 2 dx j + δ 3j dx 3 dx j (3) Summing over j index... ds 2 3D = δ 11dx 1 dx 1 + δ 12 dx 1 dx 2 + δ 13 dx 1 dx 3 = δ 21 dx 2 dx 1 + δ 22 dx 2 dx 2 + δ 23 dx 2 dx 3 = δ 31 dx 3 dx 1 + δ 32 dx 3 dx 2 + δ 33 dx 3 dx 3 (4) ds 2 3D = dx 1dx 1 + dx 2 dx 2 + dx 3 dx 3 = dx 2 + dy 2 + dz 2 (5)

6 Euclidean Example 5 Example: What is the Euclidean line element in sphericalpolar coordinates? x = r sin θ cos φ y = r sin θ sin φ z = r cos θ Taking the differentials... (6) dx = d(r sin θ cos φ) = sin θ cos φdr + r cos θ cos φdθ r sin θ sin φdφ (7) and plugging into the cartesian line element ds 2 3D = dx2 + dy 2 + dz 2 yields the spherical-polar line element ds 2 3D = dr2 + r 2 (dθ 2 + sin θ 2 dφ 2 ) Example of a coordinate transformation

7 2-Sphere Example 6 Example: What is the line element for the unit 2-sphere? locus of points in R 3 at unit distance from origin Set r = 1 and dr = 0 ds 2 2 = dθ2 + sin 2 θdφ 2 Non-Euclidean manifold

8 Special Relativity 7 Special Theory of Relativity Postulates: All inertial observers are equivalent c = constant Consequences: Time and length are relative quantities ds 2 3D is NOT invariant Lorentz transformations ds 2 3D ds 2 3D v x x x = γ(x vt) t = γ(t vx/c 2 ) (8)

9 Minkowski Spacetime 8 Line element in Minkowski space ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 = c 2 dt 2 + d x d x (9) t t dx dx ds dt dt x x ds 2 is the line element measuring length ds 2 is invariant under rotations ds 2 = ds 2 (10)

10 Minkowski Spacetime 9 Proper time: ds 2 = ds 2 (11) For proper time, set d x = 0 solving for dτ dτ = dt = dt γ c 2 dτ 2 = c 2 dt 2 + d x d x 1 v 2 /c 2 where v 2 = d x dt d x dt (12)

11 Minkowski Spacetime 10 Line element using matrix notation ds 2 = η µν dx µ dx ν where η µν = (13) Again, repeated index sum over index ds 2 = η µν dx µ dx ν = 3 3 µ=0 ν=0 η µν dx µ dx ν x µ,ν : ct x y z η µν is the Minkowski metric

12 Inertial vs. Gravitational Mass 11 Newton s Second Law F = m i a Newton s Law of Gravitation (uniform field) F g = m g φ (= m g g) Electric Field F e = QE g E m m 1 2 q 1 q 2 a = (m g /m i ) g a = (Q/m i ) E

13 Principle of Equivalence 12 Eötvös experiment verified that m g = m i to 1 part in 10 12! The Weak Equivalence Principle is g = a All objects fall at the same rate (independent of the mass). Suggests a preferred class of trajectories through spacetime. inertial (freely-falling) trajectories

14 Principle of Equivalence 13 The motion of freely-falling particles is the same in a gravitational field and a uniformly accelerated frame, in small enough regions of spacetime. What about massless particles? Elevator observer sees light moving in a curved path

15 Principle of Equivalence 14 Is Earth observer inertial? Elevator observer is non-inertial Earth observer is non-inertial! Coincidence(?) in Newtonian theory All inertial forces have the mass as a constant of proportionality in them....as does the gravitational force. Einstein suggested that we treat gravitation as an inertial effect (from not using an inertial frame).

16 Non-Euclidean Spacetime 15 In Minkowski coordinates in Special Relativity, the equation for a test particle is d 2 x µ dτ 2 = 0 For a non-inertial frame of reference, the equation becomes d 2 x µ dτ 2 + dx α dx β Γµ αβ dτ dτ = 0 the additional terms are the inertial force terms. By principle of equivalence, the gravitational forces should be given by an appropriate Γ µ αβ. Geodesic equation - Free particles move along paths of shortest possible distances

17 Non-Euclidean Spacetime 16 Line element using curved space ds 2 = g µν dx µ dx ν g µν is the metric tensor η µν is the flat limit of g µν g µν = g νµ Again, repeated index sum over index ds 2 = 3 3 µ=0 ν=0 g µν dx µ dx ν ds 2 = g 00 dx 0 dx 0 + g 01 dx 0 dx 1 + g 02 dx 0 dx g 13 dx 1 dx g 33 dx 3 dx 3 (14) g µν defines the geometry of spacetime Know g µν, Know Geometry

18 2-Sphere Example 17 Example: What is the metric tensor of the unit 2-sphere? ds 2 2 = dθ2 + sin 2 θdφ 2 The metric tensor is ( 1 0 g µν = 0 sin 2 θ ) (15) or writing the components explicitly... g θθ = 1, g θφ = g φθ = 0, g φφ = sin 2 θ What is the inverse metric? (g g 1 = 1) ( ) (g µν ) 1 = g µν 1 0 = 0 1/ sin 2 θ with components... g θθ = 1, g θφ = g φθ = 0, g φφ = 1/ sin 2 θ

19 Christoffel symbol 18 All the ways curvature manifests itself rely on something called a connection. where Γ α βγ = 1 2 gαδ ( β g γδ + γ g βδ δ g βγ ) β x β δ is a repeated index Sum! Looks like a tensor... but it s not a tensor. Notice: Γ = Γ(g, g) Christoffel is a function of the metric and the derivative of the metric

20 2-Sphere Example 19 Example: What are the possible connections for the unit 2-sphere? ds 2 2 = dθ2 + sin 2 θdφ 2 Components of the metric: Answer: g θθ = 1, g θφ = g φθ = 0, g φφ = sin 2 θ Γ θ θθ, Γθ θφ, Γφ θθ, Γθ φφ, Γφ θφ, Γφ φφ What are the connections for the unit 2-sphere? Γ θ φφ = 1 2 gθδ ( φ g φδ + φ g φδ δ g φφ ) = 1 2 gθθ ( 2 φ g φθ θ g φφ ) = 1 2 θ (sin2 θ) = sin θ cos θ Γ φ θφ = cot θ, Γθ θθ = Γθ θφ = Γφ θθ = Γφ φφ = 0 (16)

21 Einstein Field Equations 20 General Theory of Relativity G µν = R µν 1 2 g µνr = 8πGT µν G µν is the Einstein tensor describing the curvature of space. T µν is the stress-energy tensor which describes matter. MATTER tells space how to CURVE and SPACE tells matter how to MOVE!

22 Einstein Field Equations 21 Einstein Field Equations G µν = R µν 1 2 g µνr = 8πGT µν Subscripts label elements of each matrix Set of 10 second-order, non-linear, partial differential equations EM, Strong, & Weak fields on spacetime Gravity is curvature of spacetime itself!

23 Properties 22 Einstein Field Equations G µν = R µν 1 2 g µνr = 8πGT µν where R µν = α Γ α µν ν Γ α µα + Γ α µνγ β αβ Γβ µαγ α βν is the Ricci tensor and is the Ricci scalar. R = g µν R µν Notice: R µν = R µν (Γ, Γ) but Γ = Γ(g, g) The Einstein Tensor G µν = G µν (g, g, 2 g) is written entirely in terms of the metric tensor!

24 2-Sphere Example 23 Example: Line element for 2-sphere of radius a ds 2 2 = a2 (dθ 2 + sin 2 θdφ 2 ) Γ θ φφ = sin θ cos θ Γ φ θφ = cot θ 0 = Γ θ θθ = Γθ θφ = Γφ θθ = Γφ φφ (17) What are the Ricci tensors? R φφ = α Γ α φφ φγ α φα + Γα φφ Γβ αβ Γβ φα Γα βφ Summing out indicies... R φφ = θ Γ θ φφ + Γθ φφ Γβ θβ Γβ φθ Γθ βφ Γβ φφ Γφ βφ = θ Γ θ φφ + Γθ φφ Γφ θφ Γφ φθ Γθ φφ Γθ φφ Γφ θφ = θ Γ θ φφ Γθ φφ Γφ θφ (18)

25 2-Sphere Example 24 Plugging in the functions R φφ = θ ( sin θ cos θ) + cot θ sin θ cos θ = sin 2 θ (19) Likewise, R θθ = 1 What is the Ricci scalar? R = g µν R µν = g θθ R θθ + g φφ R φφ = 1 a 2 sin 2 θ sin2 θ + 1 a 2 = 2 a 2 (20) The Ricci scalar completely characterizes curvature (2D).... is constant on 2-sphere.... decreases for increasing a.

26 2-Sphere Example 25 In 1916, Karl Schwarzschild presented an EXACT solution! Choose a point mass Choose metric ansatz T µν = mδ 0 µδ 0 ν δ 3 ( x) ds 2 = e A(r) dt 2 + e B(r) d x d x Plug metric ansatz into the Einstein Field equation The solution is ds 2 = ( 1 2GM ) dt 2 + r 1 ( 1 2GM r )dr 2 + r 2 dω 2 where dω 2 = dθ 2 + sin 2 θdφ 2 is the differential solid angle.

27 Schwarzschild line element 26 ds 2 = ( 1 2GM ) dt 2 + r 1 ( 1 2GM r )dr 2 + r 2 dω 2 When r r sch = 2GM dt 2 0 & dr 2 Schwarzschild radius Not a real singularity When r 0 dt 2 & dr 2 0 Spacetime singularity! Solution predicts Black holes!