PROCESAMIENTO Y ANÁLISIS DE LAS ENCUESTAS APLICADAS

The basic idea of Einstein’s theory of gravitation is to look at the interaction among bodies in a geometrical way: each massive body modifies the space-time around it, so that any test particle (either massive or massless) travels along the geodesics in the modified

28 Mathematical description of the Universe

geometry. In a renowned summarizing motto, matter tells the space-time how to change and the space-time tells the matter how to move (Misner et al., 1973).

In this view, the determination of the motion of a test particle in a gravitational field becomes the determination of the metric describing the space-time in which the test particle is moving along the geodesics. The steps to be performed are then:

• given a mass distribution, to determine the metric tensor of the modified space-time geometry;

• computed the metric tensor, to write the equations of motion of a test particle in that geometry.

In the following sections we will address these two points using the standard relativistic formalism. We will adopt Einstein’s summation convention, and denote the components of a metric tensor with gij. By definition, the space-time interval ds will be

ds2 =gijdxidxj (1.1)

with gij =gji (symmetry condition of any metric tensor) andgij the components of the

inverse of the metric.

In a 4-dimensional space-time (3-dimensional space plus 1-dimensional time), the number of independent components is 10; often, Roman letters are used for the spatial indices running between 1 and 3 and Greek indices in general when including also the temporal 0th_{-component. The metrics we deal with are the usual relativistic metrics, so the spatial}

coordinates have a different sign than the time coordinate and the signature is (+,–,–,–).

1.1.1 The field equations

In order to determine the metric tensor, Einstein proposed the following field equations (Einstein, 1916)

Gµν =κTµν (1.2)

where Tµν are the components of the stress-energy tensor describing the distribution of

matter, or, better, of the sources of the gravitational field andGµν are the components of

the so-called Einstein tensor, related to the geometry of the space-time.

The stress-energy tensor in 4 dimensions has 16 components, but it is symmetric under reflection of coordinate system (energy does not depend on the sign of the axes),Tµν =Tνµ,

1.1 The approach of General Relativity 29

therefore the independent components are only 10. They are given by the variation of the action S with respect to the metric tensor, times a factor dependent on the speed of lightc and on the determinant of the metricg (Landau and Lifshitz, 1975)

Tµν = 2c √ −g δS δgµν or T µν ₌ −√2c −g δS δgµν (1.3)

from which the symmetry is clear. The transition from covariant to contravariant components is done via gµσ_g

σν = δµν, implying δgµν = −gµρgνσδgρσ. The expression

for Gµν is a combination of the Ricci tensor Rµν and the Ricci scalar R =gµνRµν:

Gµν ≡Rµν−

1

2gµνR. (1.4)

The Ricci tensor is defined as

Rµν =Rλµλν (1.5)

obtained contracting the contravariant component of the Riemann tensor with its second covariant component1_{. The components of the Riemann tensor depend on the convention}

adopted; we define them as:

Rλ_µνρ = Γλ_{µρ,ν −}Γλ_µν,ρ+ Γλ_σνΓσ_µρ−Γλ_σρΓσ_µν (1.6) being Γλ_µν = 1 2g λσ_(g σµ,ν +gνσ,µ−gµν,σ) (1.7)

the Christoffel coefficients and having used the comma for simple derivatives and the semicolon for covariant derivatives2_{. Given these relations, R}

µν and Gµν result to be

symmetric as well.

The expression (1.2) represents 16 equations of which only 10 are independent, because of the previous symmetry reasons. In virtue of Bianchi’s identities (Gµν

;ν = 0) and the

conservation rules (Tµν

;ν = 0), taking the quadri-divergence of (1.2), we get four identities

0 = 0, meaning that the problem is well posed.

The constantκappearing in the field equations (1.2) is calibrated on the Poisson equation

1_{There is only one possible independent contraction.}

2_{For any vector with componentsV}i_{, the covariant derivative with respect to thej-th coordinate is}

Vi

;j=Vi,j+ ΓikjVk (1.8)

and if we consider the covariant derivative forVi= gijVj, the plus sign is replaced by a minus sign. For a generic tensor

Ti1i2...

j1j2...similar rules apply:

Ti1i2... j1j2...;k=T i1i2... j1j2... ,k+Γ i1 lkT li2... j1j2...+Γ i2 lkT i1l... j1j2...+...−Γ l j1kT i1i2... lj2...−Γ l j2kT i1i2... j1l...−... . (1.9)

30 Mathematical description of the Universe

in the classical “static”, “non-relativistic”, “weak field” limit, so that the right Newtonian behaviour at low energies is retried. In practice, relations (1.2) are expanded at the first order, and reduced to the only significative relation

G00 =κT00. (1.10)

If we imposes that (1.10) coincides with the Poisson equation

△φ(x) = 4πGρ(x) (1.11)

where φ(x) is the gravitational potential and ρ(x) the density distribution at the spatial position x, this procedure leads to3 _{κ= 8πG/c}4 _{and the field equations can be rewritten}

as Rµν− 1 2gµνR = 8πG c4 Tµν. (1.12)

It is now possible to determine gµν _{from (1.12), once T}µν _{is assigned. The addition of}

a constant term like Λgµν does not change the conservation laws, as (Λgµν);ν = 0. The

constant Λ is called cosmological constant and was first introduced by Einstein in 1917: long discussions followed and still exist about its physical meaning.

1.1.2 The geodesics equations

The second step is to write down the equations of motion for a test particle moving in a space-time specified by the metric gµν_{. This is a relatively simple problem, because a test}

particle falling in a gravitational potential will travel along the geodesics, so the equations of motions are just the equations of the geodesics:

d2_xλ dτ2 + Γ λ µν dxµ dτ dxν dτ = 0 (1.13)

where xλ _{are the space-time components of the particle and τ is the variable}

parameterizing the geodesics, alias the proper time. The first term can be seen as the kinematic term, the second one as the potential term arising from the metric hidden in the Christoffel coefficients.

Also these equations lead to the correct classical limit

¨

xi =₋∂φ(x)

∂xi

(1.14)

3 _{The expression for κdepends on the conventions: sometimes ac}2 _{is included in the stress-energy tensor obtaining}

κ= 8πG/c2_{, or the Riemann tensor is defined with the opposite sign getting κ=−8πG/c}4_{. The valueκ=−8πG/c}2 _is

obtained if thec2_{is included inTµνand the Riemann tensor is defined with a different sign. In his original work, Einstein}

usedκ = 8πG/c2 _{≃ 1.87·10}−27_{dyn s}2_g−2 _{(Einstein, 1916). Anyway, for possible notations see also Weinberg (1972);}