146 3 Connections and Metrics
Riemannian manifold the metric is positive definite and there is only one type of geodesics. Indeed the norm of the tangent vector is always positive and the auxiliary condition (3.8.7) defining the affine parameter is unique. In a Lorentzian case, on the other hand, we have three kinds of geodesics depending on the sign of the norm of the tangent vector. Time-like geodesics are those where g(t, t) > 0 and the auxiliary condition is precisely stated as in (3.8.7). However we have also space-like geodesics where g(t, t) < 0 and null-like geodesics where g(t, t) = 0. In these cases the auxiliary condition defining the affine parameter is reformulated as 2L = −1 and 2L = 0, respectively.
In General Relativity, time-like geodesics are the world-lines traced in space-time by massive particles that move at a speed less than that of light. Null-like geodesics are the world-lines traced by mass-less particles moving at the speed of light, while space-like geodesics, corresponding to superluminal velocities violate causality and cannot be traveled by any physical particle.
3.9.1 The Lorentzian Example of dS2
An interesting toy example that can be used to illustrate in a pedagogical way many aspects of the so far developed theory is given by 2-dimensional de Sitter space. We can describe this pseudo-Riemannian manifold as an algebraic locus in R3, writing the following quadratic equation:
R3 AdS2 : −X2 + Y 2 + Z2 = −1 |
(3.9.1) |
A parametric solution of the defining locus equation (3.9.1) is easily obtained by the following position:
X = sinh t; Y = cosh t sin θ ; Z = cosh t cos θ |
(3.9.2) |
and an overall picture of the manifold is given in Fig. 3.19.
The parameters t and θ can be taken as coordinates on the dS2 surface on which we can define a Lorentzian metric by means of the pull-back of the standard SO(1, 2) metric on three-dimensional Minkowski space, namely:
dsdS2 2 = −dX2 + dY 2 + dZ2 |
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= −dt2 + cosh2 t dθ 2 |
(3.9.3) |
The first thing to note about the above metric is that it describes an expanding twodimensional universe where the spatial sections at constant time t = const are circles S1. Indeed the angle θ can be regarded as the coordinate on S1 and dθ 2 = dsS21 is the corresponding metric, so that we can write:
dsAdS2 |
2 = −dt2 + a2(t) dsS21 |
(3.9.4) |
where:
3.9 Geodesics in Lorentzian and Riemannian Manifolds: Two Simple Examples |
147 |
Fig. 3.19 Two-dimensional de Sitter space is a hyperbolic rotational surface that can be visualized in three dimension
a(t) = cosh t |
(3.9.5) |
The reader should remember the paradigm provided by (3.9.4) because this is precisely the structure that we are going to meet in the discussion of relativistic cosmology (see Chap. 5 of Volume 2). The second important thing to note about the metric (3.9.3) is that it has Lorentzian signature. Hence we are not supposed to find just one type of geodesics, rather we have to discuss three types of them:
1.The null geodesics for which the tangent vector is light-like.
2.The time geodesics for which the tangent vector is time-like.
3.The space geodesics for which the tangent vector is space-like.
According to our general discussion, the proper-length of any curve on dS2 is given by the value of the following integral:
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where λ is any parameter labeling the points along the curve. Performing a variational calculation we get that the length is extremal if
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Hence, as long as we use for tion
gμν dxμ dxν = −t˙2 dλ dλ
we can just treat:
λ an affine parameter, defined by the auxiliary condi-
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3 Connections and Metrics |
as the Lagrangian of an ordinary mechanical problem. The corresponding EulerLagrange equations of motion are:
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The first of the above equations shows that θ is a cyclic variable and hence we have a first integral of the motion:
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(3.9.11) |
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which deserves the name of angular momentum. Indeed the existence of this firstintegral follows from the SO(2) rotational symmetry of the metric (3.9.3), as we will show when we discuss the concept of isometries and Killing vectors. Thanks to and to the auxiliary condition (3.9.8), the geodesic equations are immediately reduced to quadratures. Let us discuss the resulting three types of geodesics separately.
3.9.1.1 Null Geodesics
For null-geodesics we have:
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Combining this information with (3.9.11) we immediately get:
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cosh2 t
(3.9.12)
(3.9.13)
The ratio of the above two equations yields the differential equation of the nullorbits:
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which is immediately integrated in the following form:
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where the arbitrary angle α is the integration constant that parameterizes the family of all possible null-like curves on AdS2. In order to visualize the structure of such
3.9 Geodesics in Lorentzian and Riemannian Manifolds: Two Simple Examples |
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Fig. 3.20 The null geodesics on dS2 are straight lines lying on the hyperbolic surface. In this figure we show a family of these straight lines parameterized by the angle α in the range {− π5 , π7 }
curves in the ambient three-dimensional space, it is convenient to use the following elliptic and hyperbolic trigonometric identities:
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Setting y |
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tions (3.9.2) and also (3.9.16), we obtain the form of the null geodesics in R3: |
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It is evident from (3.9.17) that null geodesics are straight-lines, yet straight-lines that lye on the hyperbolic dS2 surface (see Fig. 3.20).
3.9.1.2 Time-Like Geodesics |
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For time-geodesics we have: |
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t2 |
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θ 2 |
(3.9.18) |
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t ˙ |
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and following the same steps as in the previous case we obtain the following differential equation for time-like orbits
dθ |
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cosh t |
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cosh2 t |
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which is immediately reduced to quadratures and integrated as follows:
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θ + α |
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(3.9.20) |
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