When the geodesic corresponds to the path taken by a material particle, it is natural to use the proper time as a parameter for the curve. This is what was done in arriving atEq. (4.49)which can be rewritten in terms of the coordinates as
d2xi
dτ2 + Γijkdxj dτ
dxk
dτ = 0. (4.67)
But a geodesic in the spacetime, like any other curve xi(λ), is a geometrical entity that is independent of the parameter that is used to describe it. We want to comment on the occurrence of the proper time τ inEq. (4.49)and some related issues.
Note that the second equality inEq. (4.39)defining the actionA does not use any parameters. One says that the actionA is re-parametrization invariant. In proceed-ing fromEq. (4.40)toEq. (4.41) we have introduced dτ by hand which appears
4.5 Geodesic curves 157 in the final equation. SinceA is re-parametrization invariant, and the geodesic is a geometric entity that is independent of the parametrization used to define it, we are allowed to use any other parameter λ, say, to describe the same curve. If the new parameter is expressed in terms of the old by a function τ = τ (λ), we can convert the derivatives inEq. (4.67)from λ to τ and – after some simple algebra – arrive at
d2xi
This is a slightly more general form of a geodesic equation which does not restrict the parameter that is used to describe the geodesic. If τ is a linear function of λ, the right hand side vanishes and we get back the same equation as before. A class of parameters related to each other by linear transformations that preserves the form ofEq. (4.67)are called affine parameters. Proper time is an affine parameter for timelike geodesics. Incidentally, our analysis also shows that the arc length of the curve is an affine parameter for the spacelike geodesics.
Equation (4.68) can be written as ua∇aui = f (λ)ui where ua = dxa/dλ.
This equation has a simple physical interpretation. The left hand side gives the acceleration ai of the trajectory; we would expect the curve to be non-accelerating when the spatial part of the acceleration vanishes in the instantaneous comoving frame. This can happen ifa is in the direction of u which is precisely what the equation indicates.
Given any equation of this form, one can also make the inverse transformation from the parameter λ to another parameter μ, say, so that in terms of the new parameter the geodesic curve xi(μ) satisfies the equation ua∇aui = 0, where ua = dxa/dμ. The explicit form of this transformation is easily determined using Eq. (4.68):
Therefore, one can always choose an affine parameter to describe a geodesic.
It is also possible to provide several other variational principles from which the geodesic equation with an affine parametrization can be obtained. Consider a modified action principle with
This is essentially a classical mechanics problem with λ playing the role of time and xi playing the role of generalized coordinates. Variation of this action keeping xi(λ) fixed at the end points will lead to the Euler Lagrange equation
∂aL = (d/dλ)(∂L/∂ ˙xa), which becomes d
dλ
gab
dxa dλ
= 1
2(∂bgij)dxi dλ
dxj
dλ. (4.71)
This is identical to Eq. (4.42), provided λ is an affine parameter, showing that the variational principle based on A1 leads to the same equation of motion. If Q = (gab ˙xa ˙xb)1/2, where the dot denotes the derivative with respect to an affine parameter, then our original choice inA was L = Q and the choice in A1 corre-sponds to L = Q2. There is, however, one crucial difference between the original action A and any other choice. The action A, as mentioned before, is invariant under the re-parametrization τ → λ = λ(τ). It is not necessary to choose an affine parameter with A, but more generalized actions like A1 will lead to the correct geodesic equation only when the affine parameter is used as a parameter.
It is trivial to verify that if a geodesic is timelike, null or spacelike at a given event, then it will continue to remain so. This arises from the result
d
dλ(uaua) = uj∇j(uaua) = 2ua(uj∇jua) = 0, (4.72) where the last equality follows from the fact that uasatisfies the geodesic equation.
This shows that uauais a constant along the geodesic.
A null geodesic will be the path taken by a light ray in a curved spacetime just as a timelike geodesic describes the path of a material particle. The issue of the parameter that is used to describe a geodesic equation becomes particularly impor-tant when the geodesic is null. We cannot use proper time as a valid parameter for a null geodesic since it vanishes along the path of the light ray. Hence null geodesics are, in general, described using an arbitrary parameter. That is, we define the null geodesics, in general, as the integral curves of a vector field ka(x) which satisfies kb∇bka= f (x)kaand kaka= 0. However, if kais a null vector field, then μ(x)ka will also be a null vector for an arbitrary function μ(x). Using this freedom, we can make f (x) = 0 in the geodesic equation, thereby describing the null geodesic with an affine parametrization corresponding to kb∇bka = 0 and kaka = 0. It is usual to choose the parametrization such that ka is the momentum of the photon, say, which is travelling on the null geodesic – unlike the case of a timelike geodesic in which mua gives the momentum. Note that, with this convention, the energy attributed to the particle moving in the trajectory xa(λ) by an observer with four-velocity Ua will be E = −(dxa/dλ)Ua irrespective of whether the trajectory is null or timelike.
An important example in which null geodesics occur with a non-affine parameter is in the case of – what are called – conformally related spacetimes. Two metrics are called conformally related if gab= Ω2(x)gab. It is straightforward to show that
4.5 Geodesic curves 159 the Christoffel symbols for these two metrics are related by
Γabc = Γabc+ δba∂cln Ω + δca∂bln Ω− gbc∂aln Ω. (4.73) The geodesic equation changes under such a conformal transformation. In general, if xa(λ) is a geodesic in the metric gab, it will not be a geodesic in the metric gab. One exception to this rule are null geodesics, which will continue to remain as null geodesics under conformal transformations. But the null geodesic equation will now get modified to the form
d2xa This equation is in the form ofEq. (4.68) showing that xa(λ) is still a geodesic but the parameter λ is no longer an affine parameter in the new metric; but a trans-formation to a new parameter μ with dμ/dλ = Ω−2 will reduceEq. (4.74)to the geodesic equation with an affine parameter.
In the case of null geodesics in a static spacetime (with g0α = 0 and all other components independent of x0 = t), one can introduce another variational princi-ple, which is a generalization of Fermat’s principle to curved spacetime. Consider all null curves connecting two eventsP and Q in a static spacetime. Each null curve can be described by the three functions xα(t) and will take a particular amount of coordinate time Δt to go fromP to Q. We will now show that the null geodesic connecting these two events extremizes Δt. To do this, we shall change the inde-pendent variable inEq. (4.67)from the affine parameter λ to the coordinate time t by using the relation Combining this with the zeroth component of the geodesic equation
(d2t/dλ2)
(dt/dλ)2 =− 2 Γ0γ0
(dxγ/dt)
g00 (4.77)
and using the expression for Christoffel symbols in terms of the metric, one can show, after some straightforward algebra, that
Hβγ d2xγ parameter t in a three-dimensional space with metric Hαβ. It follows that the null
geodesics in a static spacetime can be obtained from the extremum principle for coordinate time
δ
dt = 0, (4.79)
which is the same as Fermat’s principle.
This result shows that light does not travel in the three-dimensional space along a path of least length but travels along a path of least time. The difference arises because the gravitational field acts like a medium with a spatially varying refrac-tive index producing an effecrefrac-tive speed of light that is different from unity. To see this explicitly, consider a special case – which arises in several practical examples including all spherically symmetric spacetimes – in which gαβ = f2(xα) δαβ so that the three-dimensional space is conformally flat. In that case, along the light path, dt = [f /
|g00|] dl, where dl2 = δαβdxαdxβ is the usual Cartesian line element. The Fermat principle is now equivalent to the statement that such a gravi-tational field acts like a medium with a refractive index n(x) = f (x)/
|g00(x)|.
In addition to the bending of light, such an effective refractive index will also lead to a time delay in the propagation of light rays. This delay, called Shapiro time-delay has been observationally verified (seeExercise 4.8).
Exercise 4.7
Non-affine parameter: an example Vary the action functional based on the Lagrangian L =
where λ is some arbitrary parameter. Show that the resulting equation has the form in Eq. (4.68)and identify f (λ).
Exercise 4.8
Refractive index of gravity We will see inChapter 7that the spacetime around a star (like the Sun) can be expressed in the form
ds2=−
where M is the mass of the star. This is called the Schwarzschild metric which we will derive inChapter 7.
(a) Show that one can convert the spatial part of the metric into a conformally flat form by making the coordinate transformation from r to ρ with
r = ρ
4.5 Geodesic curves 161 (b) Show that, in the new coordinates, the metric has an effective refractive index given by
n(x) = [1 + (GM/2|x|)]3
[1− (GM/2|x|)]. (4.83)
The slowing down of light rays due to this ‘refractive index’ has been verified using the Cassini satellite when it was in opposition to Earth. A radio wave was sent from Earth (located at the radius r1) to the satellite (located at the radius r2) and was bounced back to Earth. Compute the line integral of 2ndl along a straightline path in isotropic coordinates and show that the time delay is given by
Δt = 4GM c3 ln
4r1r2
b2
, (4.84)
where b is the impact factor of the light ray.
(c) The circumference of a circle of coordinate radius ρ, centred at the origin, in the metric inEq. (4.81)is 2πρn(ρ). Show that this circumference has a minimum value at r = 3M . What does this imply for the existence of circular null geodesics in the spacetime?
Exercise 4.9
Practice with the Christoffel symbols The purpose of this exercise is to compute the Christoffel symbols in flat two-dimensional space in polar coordinates by different methods.
(a) Write down the coordinate transformation from (x, y) to (r, θ) and the metric in both coordinates. The Christoffel symbols vanish in the Cartesian coordinates. Use the explicit transformation law of the Christoffel symbols to obtain them in the polar coordinates.
(b) From the metric in the polar coordinates, compute the Christoffel symbols by direct differentiation.
(c) The geodesics in flat two-dimensional space are just straight lines. Choosing the geodesics judiciously and knowing the geodesic equation, determine the Christoffel symbols.
Exercise 4.10
Vanishing Hamiltonians In the text, we discussed two different variational principles: one based on L1= (−gab˙xa˙xb)1/2and the other based on L2=−gab˙xa˙xbin obvious notation.
(a) Explain why both Lagrangians work under appropriate conditions. What is the general condition under which two Lagrangians L and F (L) (where F is a monotonic function) will lead to the same equations of motion?
(b) Compute the canonical momenta pa = ∂L/∂ ˙xa as well as the Hamiltonian H = pa˙xa− L for both these Lagrangians.
(c) Show that the Hamiltonian corresponding to L1vanishes identically for any trajec-tory while the Hamiltonian corresponding to L2is conserved when the equations of motion are satisfied.
(d) Explain why any Lagrangian, like L1, which is invariant under the re-parametrization of the independent variable (‘time’) will lead to vanishing Hamiltonian. What does this mean physically?
Exercise 4.11
Transformations that leave geodesics invariant Express the geodesic equation as a dif-ferential equation for xα(t). What is the most general transformation of the Christoffel symbols that will leave these equations invariant?