Undulating Gravity (Gravitational Waves)

R(τem) := H(τem)R(τem) (2.33)

Here we have replaced the periods T by ν⁻¹ and have assumed that the difference of the τ labels is small. Speed of light being 1, this difference equals the physical distance between the galaxies.

Thus, in Robertson–Walker space-times, there is a frequency shift in the light exchanged between two galaxies thanks to the time dependent scale factor and the shift is proportional to the physical separation between them.

This matches naturally with the famous conclusion drawn by Hubble from observation of spectra from distant galaxies. He consistently observed a red-shift which was also proportional to their distance from us, now known as the Hubble Law. Putting the two equations (2.31, 2.33) together, he concluded that our universe must be expanding.

2.6 Undulating Gravity (Gravitational Waves)

Consider a small ripple of geometry on the background of the Minkowski space-time. By this we mean a metric of the form gµν(x) := ηµν + hµν(x) where h is treated as a first order quantity i.e. raising/lowering of indices is done by the background metric and only the leading, non-vanishing terms are kept. The coordinates denote the standard Cartesian coordinates of the Minkowski space-time so that the background metric takes the form η_µν :=

diag(−1, 1, 1, 1). As an example, we take,

h_µν(x) = _µν(k)e^ik·x+ ¯_µνe^−ik·x, (2.34) k²:= k^µk^νηµν= 0, µνη^µν = 0, k^µµν = 0;

The conditions on the µν and k²= 0 are equivalent to hµν(x) satisfying the equations: η^µν∂_µ∂_νh_αβ = 0 , h_µνη^µν = 0 , and ∂_µh^µν = 0. The h_µν(x) thus represents a plane wave propagating along a direction ~k with frequency ω := |k0| = |~k| and the (complex) polarization tensor, µν satisfying those conditions. The polarization tensor is said to be transverse, traceless.

It is easy to verify that these conditions do not determine the polarization tensor completely: two sets satisfying the conditions can differ as, ⁰_µν− µν = ik_µζ_ν+ ik_νζ_µ, k · ζ = 0. The 10 components of the polarization tensor satisfy 5 conditions and have a further freedom worth 3 parameters (since k · ζ = 0).

This leaves two independent components of polarization tensor.

Given the wave 4-vector k, we can define a set of 4, complex, linearly independent null (light-like) vectors: k, `, m, ¯m with non-zero scalar products

k · ` = −1, m · ¯m = 1. Using this basis of null vectors, we can write µν :=

Φabe^a_µe^b_ν , {e^a} = (k, `, m, ¯m). Using the 5 conditions on the polarization tensor as well as the exploiting the 3 parameter freedom, all Φabcan be chosen to be zero except Φmm, Φm ¯¯m. We will thus work with,

h_µν(x) = e^ik·x(Φ_mmm_µm_ν+ Φ_{m ¯¯m}m¯_µm¯_ν)

Following properties are immediately verified: (a) these waves are trans-verse i.e. the polarization tensor is orthogonal to the wave vector (k · m = 0 = k · ¯m); (b) if we perform a rotation through an angle θ, in the plane transverse to ~k, then the Φmm amplitude changes by e^−2iθ while the other amplitude changes by e^2iθ. The waves are then said to have helicities ±2. The Riemann–Christoffel connection and the Riemann tensor, to first order in h, are given by,

Γ^λ_µν = i

2e^ik·x kµ^λ_ν+ kν^λ_µ− k^λµν
+ Complex Conjugate (2.35) R^α_λµν = 1

2e^ik·xk^α(λνkµ− λµkν) − kλ(^α_νkµ− ^α_µkν) + C.C. (2.36) The see how such a wave may affect test particles, we can use the geodesic deviation equation. The relative acceleration between two neighboring time-like geodesics is given by (14.11),

a^α = − R^α_λµνu^λX^µu^ν

where, u^λ is a reference geodesic while X^µ is a deviation vector (u · X = 0) to a nearby geodesic. In the rest frame of the reference geodesic, u⁰ = 1/√

−g00, uⁱ= 0, we get u · m = 0 = u · ¯m. Substituting for our polarization tensors and using ω := k · u, the frequency of the wave as measured by the freely falling observer u, we get,

a^α = −1

2ω²{Φmm(m · X)m^α+ Φm ¯¯m( ¯m · X) ¯m^α} e^ik·x+ C.C.

Notice that the relative acceleration is in the plane transverse to the wave vector, ~k. Furthermore, if a deviation vector is along ~k, then there is no tidal acceleration. Taking a ring of test masses in the plane perpendicular to the direction of the wave, one can develop a detailed picture of how these masses respond to a passing gravitational wave [8].

This is our second example of a time varying geometry. It describes a ripple of curvature. It has a characteristic effect on test bodies implied by its helicity 2 nature.

Chapter 3

Dynamics in Space-Time

In this chapter, we will consider motions of test objects i.e. objects which will carry mass, energy etc. whose influence on the space-time geometry however can be neglected. We take as given, some space-time and study some generic properties of motion of point particles, small rotating objects and wave motion.

These equations are generically obtained by appealing to Principle of general covariance and Principle of equivalence.

3.1 Particle Motion Including Spin

Having gotten a more general framework for a space-time, a natural ques-tion is: how are the (classical) non-gravitaques-tional laws of physics to be adapted to this generalized framework?. The laws of physics here, refer to the laws of point particle mechanics and laws of dynamics of fields. We already made such an adaptation while going from the Newtonian framework to the special relativistic one: the kinematical quantities associated with a particle e.g. its position, velocity, acceleration are to be described as 4-vectors, its intrinsic attributes are to be the rest-mass (for a massive particle) - a scalar, its intrin-sic angular momentum - a space-like 4-vector and force to be a 4-vector. The Newton’s laws of motion are to be expressed as a Lorentz-covariant equations.

In going to the general relativistic adaptation, the Lorentz tensors are pro-moted to general tensors and so are the equations. The position of a particle ceases to be a vector. Wherever there are derivatives, these are to be replaced by covariant derivatives. This procedure is taken as a statement of Princi-ple of General Covariance. Since the covariant derivatives reduce to ordinary derivatives in a locally inertial coordinate system, the procedure amounts to stipulating that Laws of physics should take the special relativistic form in a lo-cal inertial coordinate system. This formulation is sometimes referred to as the Principle of Equivalence. By contrast, the assumed equality of gravitational and inertial masses is referred to as the weak principle of equivalence. There are caveats to the procedure of replacing coordinate derivatives by covariant derivatives when higher order derivatives are involved. These stem from the fact that while coordinate derivatives commute, the covariant derivatives do 29

not and different orderings differ by curvature terms via the Ricci identity.

More on this below.

Free Motion: In special relativity (and in Newtonian theory too), all free particles follow straight line trajectories (world lines) regardless of their non-zero mass i.e. in a locally inertial coordinate system, its trajectory satisfies the equation: ^d_dτ^2x2^µ = 0. This can be manipulated as,

0 = d²x^µ

dτ² = v^ν∂νv^µ (v^µ :=dx^µ

dτ is used)

= v^ν∇_νv^µ (general covariance)

= v^ν∂νv^µ+ v^νΓ^µ_νλX^λ (definition of covariant derivative)

= d²x^µ

dτ² + Γ^µ_νλdx^ν dτ

dx^λ

dτ (the geodesic equation!). (3.1) Thus a trajectory of a free particle is described by a geodesic.

In special relativity, the invariant interval on a world line of a massive particle is time-like while that on a massless particle (light) is light-like or null. This is equivalent to their velocities being time-like or null vectors. The same distinction holds for general space-time by general covariance. Since the norm of velocity, gµνv^µv^ν is constant along a geodesic, the entire geodesic can be labelled as being time-like or null. Thus, both massive and massless, free particles follow time-like (respectively null) geodesics in any space-time.

What if a particle has an ‘intrinsic spin’ ? This would be a model for a small gyroscope having spin angular momentum. In special relativity, such a spin will be described by a 4-vector which is space-like. This is because, in the rest frame of a particle (necessarily massive)¹, the spin is a 3-vector pointing in some direction and having some magnitude √

~

s · ~s. Its 4-norm is positive and hence space-like. The velocity in the same frame is a time-like vector with no spatial component i.e. ηµνv^µS^ν = 0. In a torque-free, free motion, this 4-vector will be preserved along the time-like geodesic: ^dS_dτ^µ = 0, which generalizes into the equation,

dS^µ

dτ = 0 = v^ν∂νS^µ+ Γ^µ_νλv^νS^λ, gµνv^µS^ν = 0, s²:= gµνS^µS^ν. (3.2) For a non-free motion with a force being F^µ, the equations of motion for a single, non-spinning particle then take the form,

m₀v^ν∇νv^µ = F^µ, g_µνF^µv^ν = 0 ; (3.3) An example of such a force, for a charged particle, is the Lorentz force F^µ := qF^µνvν. Another example would be F^µ ∼ A^µαβvαvβ where the third rank tensor A, is symmetric in the last two indices and anti-symmetric in the first two indices.

1For a massless particle, there is no notion of a spin, but one has the notion of helicity.

Dynamics in Space-Time 31 If the body is spinning as well, then the force may act with/without gen-erating a torque. In such a case of torque-free, accelerated motion of a small spinning body, we write ^dS_dτ^µ = ξv^µsince in the rest frame, the spin should not change its direction. Preservation of S · v = 0 then determines, ξ = S · F /m⁰ and we get,

v^ν∇νS^µ = (Sαv · ∇v^α) v^µ =  S · F m0



v^µ. (3.4)

Thus under torque-free, accelerated motion, the spin vector satisfies an equa-tion (the first equality) known as the Fermi Transport Equaequa-tion. For geodesic motion, (F^µ = 0) it reduces to the parallel transport equation for the spin vector.

For a small spinning body or an idealized point spin, we may have only torque-free motion.

Even for the free fall motion, we should appreciate that the spin vector will ‘precess’ in general even though it is non-precessing in the rest frame.

This precession - or change of direction of the spin - is defined relative to some fixed direction defined by a distant star or quasar. This can be computed by solving the parallel transport equation for S^µ [2] and is sensitive to the curvature² (‘geodetic precession/De Sitter precession’) as well as the spin of the rotating body (‘frame dragging effect/Lense-Thirring effect’) warping the space-time geometry. An experiment to detect these precessions in the near Earth geometry, thereby testing general relativity was proposed by Pugh and Schiff in 1959 [9] and was realized some 45 years later by the Gravity Probe B mission [10].

For an extended body though, a torque will in general be induced due to the differential forces on parts of the extended body and these can be obtained from the deviation equation (14.11). For instance, even though Earth’s motion around the Sun may be well approximated as a free fall (geodesic), there is a torque induced on the Earth’s spin by the tidal forces causing precession of the equinoxes [8]. For analysis of general motion of an extended body, please see [11–15].