Lecture: Lorentz Invariant Dynamics

Transcription

1 Chapter 5 Lecture: Lorentz Invariant Dynamics In the preceding chapter we introduced the Minkowski metric and covariance with respect to Lorentz transformations between inertial systems. This was shown to lead to the basic properties of special relativity: relativity of simultaneity, time dilation, and space contraction. In this chapter we continue that discussion for flat Minkowski space and consider general properties of trajectories for particles and for light in Minkowski spacetime. 107

2 108 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS Geodesics A metric allows us to define geodesics: A geodesic for a space is a path that represents the shortest distance between any two points. A geodesic may also be viewed as the straightest possible path between two points. More technically, a geodesic is a curve that parallel-transports its own tangent vector. FLAT SPACE: The shortest distance between two points is a straight line. Thus, the geodesics in Euclidean space are given by r = 0 (Newton s 1st law) MINKOWSKI SPACE: d 2 t dτ 2 = 0 d 2 r dτ 2 = 0, where τ is the proper time (the time that would be measured by a clock carried along a worldline). In both cases, the geodesics are straight lines (generally will not be true in curved spacetime).

3 Geometrized Units It is convenient to introduce a new set of units in which c and/or G can be set to unit value so that they do not appear explicitly in equations. These are called geometrized units or c = G = 1 units. Geometrized units, and how to convert between standard units and geometrized units, are explained in examples below and in an Appendix.

4 110 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS Assuming c = G = 1 and setting 1 = c = cm s 1 1 = G = cm 3 g 1 s 2, we may solve for standard units like seconds in terms of these new units. For example, from the first equation 1 s = cm and from the second 1 g = cm 3 s 2 ( ) = cm = cm. cm So both time and mass have the dimension of length in geometrized units. Likewise, we may derive from the above relations 1 erg = 1 g cm 2 s 2 = cm 1 g cm 3 = cm 2 1M = km, and so on. Velocity is dimensionless in these units since 1 cm s 1 = (that is, v is measured in units of v/c). From this point onward, we shall commonly work in geometrized units unless the explicit restoration of c or G factors in an equation is desirable for clarity or to make a particular point.

5 Velocities Particles with finite mass follow timelike worldlines. The worldline for a particle is conveniently parameterized in terms of a variable that changes continuously along the worldline. For timelike trajectories the natural choice for this parameter is the proper time τ. The equation of the worldline may then be expressed as x µ = x µ (τ) and we may define a velocity 4-vector (the 4-velocity) by u µ = ( dx 0 ) dτ, dx1 dτ, dx2 dτ, dx3. dτ The proper time interval dτ is related to the spacetime interval ds dτ 2 = ds 2, and the coordinate time interval dt and the proper time interval dτ are related through special-relativistic time dilation: ( dτ = dt 1 v 2) 1/2 1 ( = γ dt γ 1 v 2) 1/2 (Lorentz γ) where v is the 3-velocity, v i = dx i /dt. (c = 1 units! This would read dτ = dt(1 v 2 /c 2 ) 1/2 in standard units.)

6 112 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS Figure 5.1: The 4-velocity along a timelike worldline. The 4-velocity is tangent to the worldline of a particle at any point and lies within the forward light cone (Fig. 5.1). Since dt = γdτ, u 0 = dx0 dτ = dt dτ = γdτ dτ = γ = ( 1 v 2) 1/2 u i = dxi dτ = dxi dt }{{} dt }{{} dτ v i γ = v i γ = v i ( 1 v 2) 1/2 so that we may write for the components of the 4-velocity u µ = (γ,γv) γ = ( 1 v 2) 1/2.

7 113 Since we have ds 2 = dτ 2 = η µν dx µ x ν, which gives, upon dividing by dτ 2, 1 = η µν dx µ dτ dx ν dτ = u u, the scalar product of u with itself gives the normalization u u = η µν u µ u ν = η µν dx µ dτ dx ν dτ = 1, u µ = (γ,γv) η µν = For massive particles we may always invoke the condition u u = 1.

8 114 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS Momenta We may define the 4-momentum by p µ (E, p) = mu µ, where m is the rest mass. Since u u = 1, the normalization of the 4-momentum is p 2 p p = m 2 u u = m 2. Because u µ = (γ,γv), the components of the 4-momentum are p µ = (E, p) = (γm,γmv) p µ = η µν p ν = ( E, p), with γ = ( 1 v 2) 1/2. Thus, p 2 = m 2 implies that ( ) E p µ p µ = ( E, p) = m 2 E = p 2 + m 2, p which is just the familiar Einstein relation E = p 2 c 2 + m 2 c 4 E = mc 2 (p 0), written in c = 1 units.

9 115 ct B A x Figure 5.2: Extremizing the proper time to determine the geodesic for a particle Principle of Extremal Proper Time Principle of extremal proper time: the worldline for free particles between timelike separated points extremizes the proper time between them (Fig. 5.2). From (c = 1 units) dτ 2 = ds 2 = ( dt 2 + dx 2 + dy 2 + dz 2,) the proper time between the points A and B is τ AB = B A dτ = B A (dt 2 dx 2 dy 2 dz 2 ) 1/2

10 116 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS We may parameterize the path by a variable σ that varies continuously from 0 to 1 as the particle moves from A to B and τ AB = 1 0 [ ( dt dσ ) 2 ( dx dσ ) 2 ( dy dσ ) 2 ( dz dσ ) 2 ] 1/2 dσ The condition for an extremum is that δ dτ = 0, where the variation is generally of the form Defining a Lagrangian δ f = f x µ δxµ dx L ( η µ dx ν ) 1/2 1 µν τ AB = Ldσ dσ dσ 0 the variation δ dτ = 0 then implies the Euler Lagrange equation of motion d ( dσ L (dx µ /dσ) ) + L x µ = 0.

11 117 EXAMPLE: Consider x µ = x 1. The Euler Lagrange equation is d ( ) L dσ (dx µ + L /dσ) x }{{} µ = 0 }{{} Derivatives Coordinates For constant η µν the Lagrangian L does not depend on x 1 and the Euler Lagrange equation reduces to d ( L dσ (dx µ /dσ) ) + L x µ }{{} =0 = 0 d ( 1 dσ L dx 1 ) = 0. dσ Inserting 1/L = dσ/dτ and multiplying by dσ/dτ, gives d 2 x 1 dτ 2 = 0 Applying similar steps to the other terms then gives the general result (Exercise) d 2 x µ = 0 No curvature for geodesic dτ2 The principle of extremal proper time implies that geodesics in Minkowski space are straight lines.

12 118 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS Principle of Extremal Proper Time (Taylor and Wheeler): Spacetime shouts Go straight! The free stone obeys.... The stone s wristwatch verifies that its path is straight.

13 Light Rays For particles moving at lightspeed the rest mass is identically zero. Photons move on the light cone with the proper time between two points given by dτ 2 = ds 2 = 0, Thus photons travel any distance in zero proper time. the proper time τ is not a useful parameterization for the world line of photons and other massless particles. However, notice that we may write the curve x = t (corresponding to v = c expressed in c = 1 units) parametrically as x µ = u µ λ where u µ = (1,1,0,0) is a tangent 4-vector, and λ is a parameter. u µ = dxµ dλ

14 120 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS With this choice of parameterization the equation of motion for the light ray may be put into the same form as that for a massive particle du dλ = 0 which is analogous to Newton s first law. Parameters λ for which this is true are termed affine parameters. Affine parameters generally are not unique; for example, if λ is an affine parameter then λ multiplied by any constant is also an affine parameter. Affine parameters are convenient for light rays because they lead to equations of motion that mimic those for timelike particle trajectories.

15 121 For massive particles u u = 1, but since for this photon case u µ = (1,1,0,0) we have u µ = η µν u ν = ( 1,1,0,0) Thus for photons u u = u µ u µ = ( 1,1,0,0) = 1+1 = 0. The primary differences between equations governing the motion of massive particles and those governing the motion of massless particles (e.g., photons) in gravitational fields will be associated with the difference in 4-velocity normalizations u u = 1 (massive particles) u u = 0 (massless particles) Otherwise their equations of motion will be similar.

16 122 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS For photons we have that the energy E and momentum p are given by E = hω p = hk, where h is Planck s constant and k is the wavevector. Thus, p µ = (E, p) = ( hω, hk) = hk µ = h(ω, k). Since photons are massless, the 4-momentum and 4-wavevector are normalized such that p p = k k = 0. (which is E = pc in c = 1 units). The equations of motion for photons may also be expressed in terms of the 4-momentum or 4-wavevector, dp dλ = 0 dk dλ = 0, where λ is an affine parameter.

17 123 ct e0 e2 e1 y x Figure 5.3: Unit vectors of a local coordinate system at a point on an observer s worldline for two space and one time dimension Observers An observer moving through spacetime may be thought of as occupying a local laboratory moving on a (timelike) worldline. The observer carries four orthogonal unit vectors e 0, e 1, e 2, and e 3 that specify a local, orthonormal coordinate system (Fig. 5.3). This coordinate system defines (locally) a time direction and three space directions to which the observer will reference all measurements.

18 124 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS The timelike component e 0 will be tangent to the observer s worldline (the observer s clock is moving in that direction if it is at rest in the laboratory). Since the 4-velocity u obs of the observer is a unit tangent vector (u u = 1), we have that e 0 = u obs. and the observer may choose any mutually orthogonal set of three unit spatial vectors to complete the set, as long as they are orthogonal to e 0. Observers refer observations to the axes of their lab and its clocks. Thus, they measure components of 4-vectors along their chosen basis vectors. These components may be computed by taking scalar products with the orthonormal basis 4-vectors. Example: For the 4-momentum p = p µ e µ. We have in particular that the energy measured by an observer with 4-velocity u obs is given by since e 0 = u obs. E = p 0 = p e 0 = p u obs,

19 5.1. ISOMETRIES AND KILLING VECTORS Isometries and Killing Vectors In differential geometry, Killing vectors are standard tools for analyzing symmetries such as those that arise as conservation laws in the usual Lagrangian or Hamiltonian formulations of mechanics. In all spacetimes, whether flat or not, one constant of motion may be deduced from the normalization of the 4-velocity u µ = dx µ /dτ g µν u µ u ν = 1, corresponding to the preservation of u u. If there are additional constants of motion, they must arise from specific symmetries in the problem. In ordinary mechanics, continuous symmetries imply conservation laws. Example: conservation of angular momentum follows from a potential that is spherically symmetric. If a spacetime metric has a symmetry (termed an isometry), that too will generally imply that some quantity is conserved.

20 126 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS Suppose the metric is independent of one of the spacetime coordinates, say x 0, such that x 0 x 0 + constant leaves the metric unchanged. For such an isometry we define a unit vector pointing along the direction in which the metric is constant, ξ µ = (1,0,0,0). The vector ξ µ is termed the Killing vector associated with the symmetry. EXAMPLE: In flat space ds 2 = dx 2 + dy 2 + dz 2 and conservation of the components of linear momentum is associated with three Killing vectors (1,0,0) (0,1,0) (0,0,1) indicating invariance under translations in the x, y, and z directions, respectively.

21 5.1. ISOMETRIES AND KILLING VECTORS 127 Symmetries implied by Killing vectors mean that some quantity is conserved along a geodesic. This quantity may be found using the principle of extremal proper time (Euler Lagrange equation).

22 128 CHAPTER 5. LECTURE: LORENTZ INVARIANT DYNAMICS Example: Suppose that the metric is independent of the coordinate x 1, corresponding to a Killing vector ξ α = (0,1,0,0) Then L/ x 1 = 0 and L (dx 1 /dσ) = g 1µ dx µ L dσ = g αµξ α u µ = ξ u, where we have used dx L = ( g µ dx ν ) 1/2 µν dσ dσ Ldσ = dτ g 1µ = g αµξ α. Then the Euler Lagrange equation d ( ) L dσ (dx µ + L /dσ) x µ = 0 reduces to d (ξ u) = 0 ξ u conserved on geodesic dσ ξ u is conserved along a geodesic if ξ is a Killing vector associated with a symmetry of the spacetime metric and u is the 4-velocity.