Testing alternative theories of gravity with the BepiColombo Radio Science Experiment

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Master of Science in Mathematics

16-12-2016

M.Sc. Thesis

Testing alternative theories of gravity with the

BepiColombo Radio Science Experiment

Vincenzo Di Pierri

Supervisor:

Dr. Giacomo Tommei

Examiner:

Dr. Giulio Bau

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Introduction

General Relativity (GR) is one of the most beautiful and elegant physics theory. It was devel-oped by Albert Einstein at the beginning of ’900 with the attempt of unifying gravitation with Special Relativity (SR). SR was another Einstein’s intuition, introduced in 1905 with the aim of explaining some results of electromagnetism that seemed to enter in conflict with classical physics. In this theory space and time are not different entities, but live together in the so called space-time. The concept of absolute time was deleted, everything is relative. With GR, Einstein found a very brilliant way to keep together gravitation and electromagnetism by in-troducing some new concepts: curved space-time and the description of gravity as a geometric property of space and time. In GR, space-time is a manifold equipped with a metric, given by the so called metric tensor. Such tensor is the main character of the theory: everything moves following metric tensor rules, that is bodies move along geodesics of the metric. That is why GR belongs to the class of metric theories of gravity. Just from the beginning, GR succeeded in explaining some effects that classical physics can not explain. These effects are known as the four classical tests of GR: Mercury’s perihelion shift, deflection of the light, light-time delay and gravitational redshift.

At the moment, predictions of GR have been confirmed in all the observations and exper-iments performed (the last confirmation has been the detection of gravitational waves), but the testing of the theory is running yet. Just from the born of GR, many alternatives theories of gravity were introduced in competition to Einstein’s theory. Among them, an important class is the one admitting a metric tensor as the GR (usually coupled with other fields). Ex-periments are the best way to compare different metric theories and then to test the validity of GR. Most of the experiments are performed in the Solar System, using tracking of objects with radio waves. The Solar System region presents a weak gravitational field (the mass of the biggest planet, Jupiter, is about 1/1000 of the mass of Sun and our star has a mass many order of magnitude less than a black hole for example) and the velocities of the objects are small with respect to the velocity of light. Thus, in order to make computations in this regime, an approximation of gravitational theories has been developed: this approximation is called Post-Newtonian (PN) limit. In the second half of 20th century, a global metric theory was developed: the Parameterized Post-Newtonian (PPN) formalism. This theory gives a general form of PN metric tensor depending upon ten parameters and every single metric theory has its own set of values for these ones. The introduction of the PPN Formalism makes possible to compare different metric theories by giving a limitation of the values of the parameters through observations and measurements.

In the ’60s some attempts to unify gravitation with quantum mechanics were done and in this contest the so called torsion theories were introduced. Such theories were put apart because there were not experimental evidences. Recently, some papers about testing in the Solar System of metric theories with torsion appear. In this thesis work, following [March et al. 2011], we

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present torsion theories in their PPN expansion, with the awareness that this is just a toy model. In fact, if torsion is related to the spin of elementary particles and in macroscopic matter it is usually oriented in random way, the effect of torsion generated by massive bodies will be negligible. Following the works of [March et al. 2011] and [Mao et al. 2007] that give an adequate model for the description of torsion effects around macroscopic bodies, we derived PPN equations of motion of a test body in a space-time with torsion. We implemented such equations in the software ORBIT14, that is a software developed by the Celestial Mechanics Group of the University of Pisa in the last nine years. The software is an Orbit Determination (OD) one to use in the context of BepiColombo, a joint mission of the European Space Agency (ESA) and the Japan Aerospace eXploration Agency (JAXA) to the planet Mercury, and Juno, a NASA mission to Jupiter. It enables the generation of the simulated observables and the determination of parameters (not only the orbital ones) by means of a global least squares fit. Our work has been developed for the BepiColombo mission, that will perform a relativity experiment using a radio wave tracking.

Concerning the contents of the thesis, in Chapter 1 we introduce the basic principle of GR, seeing some basis of the so called experimental gravity.

In Chapter 2 we see how to find the Post-Newtonian metric, giving just a summary view of the main passages. Then we report the classical method to compute Post-Newtonian limit of any metric theory and we see how to calculate it for GR. In the second part we introduce the torsion tensor and we describe a general torsion theory in the Post-Newtonian framework with the help of three torsion parameters t1, t2, t3. In this formulation we compute the equations of

motion of a test body in a field generated by a massive body.

Finally, in Chapter 3 we briefly introduce the BepiColombo mission and the software OR-BIT14, describing the algorithms implemented in the software. Then we report the results of a full set of simulations of the relativity experiment and in particular the simulations made to give an estimation of the torsion parameters.

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Chapter 1

General Relativity and the four

classical tests

1.1 From Special Relativity to General Relativity

Special Relativity (SR) is a theory regarding the relationship between space and time. It was originally proposed in 1905 by Albert Einstein in order to solve the inconsistency of Newto-nian mechanics with Maxwell’s equations of electromagnetism. SR is based on following two postulates:

1) the laws of physics are invariant in all inertial systems;

2) the velocity of propagation of interactions is an universal constant, that is the speed of light c = 2.998 · 1010cm/s, and this velocity can not be exceeded..

Starting from these two postulates, Einstein built the whole theory and published it in 1905 in the paper ”On the Electrodynamics of Moving Bodies”. The main change from the Newton’s point of view is that space and time are not different entities, but they live together in the so called space-time. SR replaced the conventional notion of an absolute time with the notion of a time that depends on the reference frame and the spatial position. Lots of the consequences of SR, such as length contraction, time dilation, relativistic mass, mass-energy equivalence, universal speed limit and relativity of simultaneity, have been experimentally verified. Math-ematically, the space-time is modeled like a real manifold of dimension 4 called Minkowsky’s space-time: the first coordinate is the temporal one multiplied by the speed of light c, in order to be measured like a spatial coordinate, the latter three are the spatial ones, and every point (ct, x, y, z) is called an event. The manifold is equipped with a scalar product on the tangent space, which is defined by the Minkowsky metric tensor. The distance between two events of coordinates P1 = (ct1, x1, y1, z1) and P2 = (ct2, x2, y2, z2) is defined by

d(P1, P2) =

p c2_(t

1− t2)2− (x1− x2)2− (y1− y2)2− (z1− z2)2.

Thus we can express the line element as

ds2 = c2dt2− dx2− dy2− dz2 (1.1) The line element (1.1) can be expressed as

ds2 = ηµνdxµdxν

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using Einstein summation convention (i.e. when an index variable appears twice in a single term and is not otherwise defined, it implies summation of that term over all the values of the index), and where we have dx0= cdt with the convention that when a greek letter appear the sum is extended to all the four coordinates, when a latin letter appear the sum involve only three spatial coordinates.

The metric tensor ηµν can be represented by the matrix

ηµν =     1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1    

Thus the space-time is a metric space, with the metric that does not depend upon coordinates. We say that Minkowsky’s space-time is a flat space. If we put some event (ct, x, y, z) in the origin O of space-time, every event contained in the light cone defined by the equation

C = {P ∈ R4 | d(P, O) ≥ 0}

is related to O by causality effect (spacelike events). The events contained in Cc are related to O by contemporaneity effect (timelike events).

In order to simplify the notation, from now on we will use the geometric unit system, a system of natural units where the speed of light in vacuum c and the gravitational constant G are fixed to the value 1:

c = 1 G = 1.

In Chapter 3, where we will deal with the simulations of a radio science experiment, we will come back to the usual units.

Transformations of space-time that permit the change from an inertial reference frame to another one are the Lorentz transformations, which are the isometries of Minkowsky space-time that preserve the quadratic form

Q : R4 −→ R

(t, x, y, z) 7→ t2− x2_{− y}2_{− z}2_. (1.2)

Maxwell’s equations are invariant respect to Lorentz transformations. An example of Lorentz transformation is this: suppose S and S0 be two inertial reference frames with S0 that moves with speed v along x-axis respect to S.

The coordinates of the two systems are related by the transformation                t0 = √t − vx 1 − v2 x0 = √x − vt 1 − v2 y0= y z0 = z

Although SR agreed with Maxwell’s equations and it was confirmed by few experiments, it was in contrast with Galileo’s relative principle, that agreed with Newton’s gravitational law. The main argument contradicting the two theories (i.e. SR and Newton’s gravitational law), is the

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1.1. FROM SPECIAL RELATIVITY TO GENERAL RELATIVITY 9

speed of propagation of the gravitational interaction. Indeed, for Newton the gravitational fields propagate instantly in every point of the space, whereas for SR nothing can exceed the speed of light in vacuum. Thus, in order to put together Newton’s laws and SR in an unified theory, from 1905 to 1915 Einstein produced one of the most beautiful and elegant physics theory: General Relativity (GR). In GR gravitational attraction is seen as a modification of the space-time. The concept that plays a fundamental role in GR is the Principle of Equivalence, also known as Newton’s Equivalence Principle (NEP)

Definition 1 (NEP).

The inertial mass of any body, that is the quantity that regulates its response to an applied force, is equal to its weight, that is the quantity that regulates its response to a gravitational field:

mI = mG.

An alternative assertion of this principle is that all bodies fall in a gravitational field with the same acceleration independently from their mass. Using NEP, Einstein’s idea has been the following: if every body falls with the same acceleration in an external gravitational field, then for an observer in a freely falling reference frame in the same gravitational field the body should seem unaccelerated. This is known as Weak Equivalence Principle (WEP).

Definition 2 (WEP).

It is impossible to know if some external force is due to a gravitational field or to a non-inertial reference frame.

Thus to the extent that their mechanical motions are concerned, the bodies will behave as if gravity were absent. Einstein proposed that not only mechanical laws should behave as if gravity were absent but also other physical law should, including the laws of electromagnetism. This principle is now called Einstein Equivalence Principle (EEP).

Definition 3 (EEP). The following holds true:

(i) WEP is valid;

(ii) the outcome of any local nongravitational test experiment is independent of the velocity of the freely falling apparatus;

(iii) the outcome of any local nongravitational test experiment is independent of where and when in the universe it is performed.

As we will see in the next sections EEP leads to an explanation of gravitation as a curved space-time phenomenon. Thus in some way GR is based on a sort of equivalence between gravitational fields and non-inertial reference frames, i.e. accelerated reference frames. But these systems can not be considered equivalent at all: indeed at infinity gravitational fields go to zero, fields equivalent to non-inertial reference frames do not go to zero. In general it is possible to make a change of coordinates in order to eliminate the field when we are in presence of a non-inertial reference frame passing to an inertial one. When we have a gravitational field it is not possible to eliminate the field everywhere by a change of coordinates, but we can do it for just an infinitesimal part of the space-time.

Mathematically, as in SR, in GR space-time is modeled as a smooth connected real manifold M of dimension four, the first coordinate being the temporal one and the latter three the spatial

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ones. It has a metric structure, i.e. it has a symmetric bilinear form g defined in the tangent space, with signature (1, 3), also called Lorentzian signature. Each point of the space-time is contained in a coordinate chart, by no means unique, and this chart can be viewed as a local representation of space-time around the observer, represented by the point. The main difference between SR and GR in the mathematical structure of the space-time is that for GR the metric g depends on coordinates, thus in different points it assumes different values. As Lorentz transformations preserve the bilinear form Q given by (1.2), we want to find a group of transformations preserving the bilinear form g.

Driven by WEP, Einstein wanted to extend Lorentz transformations with general transfor-mations to pass from a generic reference frame to another one. This idea was supported by the fact that coordinates does not exist in nature, they are just a human invention, so the physical laws must be the same in every frame of reference, inertial or non-inertial. This leads to the following definition.

Definition 4 (Covariant law). Let Θ be the expression of a law. We say that Θ is a covariant law with respect to the group of transformations Γ if after any transformation of Γ, Θ still holds.

So we can reformulate (EEP) as

Definition 5 (Principle of General Covariance). Physical laws must be covariant laws under arbitrary differentiable coordinate transformations, i.e. they can not depends on coordinates. In an inertial frame of reference physical laws must reduce to the laws of SR.

Thus, instead of considering only Lorentz transformations, we consider diffeomorphisms from M to itself written as

f : M −→ M

x0µ 7→ xµ= fµ(x0ν) (1.3) with fµdifferentiable functions. Sometimes functions f are called generalized functions. Func-tions f are simply changes of basis, i.e. of coordinates, on the manifold M preserving the metric. The set of diffeomorphisms f : M → M that preserve the metric forms a group with operation given by composition. We denote this group with H.

We now have to focus on the formalism used in (1.3), where the exponent µ appears: if in a manifold M there is a bilinear form TPM × TPM → R, fixed a basis B = (ei)i∈I and

the correspondent dual basis B∗ = (ei)i∈I, the difference between covariant and contravariant

representation of a vector is given by the fact that ∀v ∈ TPM ∃!α ∈ (TPM )∗, that is the dual

space of the tangent space called cotangent space, such that α(w) = g(v, w) ∀w ∈ T_PM

so if we write α and v in coordinate respect to the basis B∗ and B, we say that α is the covariant version of v, and v is the contravariant version of α. The way to pass from covariant to contravariant and vice versa is using the matrix of the bilinear form and its inverse. For example if we set α = (α1, α2, . . . , αn) and v = (v1, v2, . . . , vn) we have

αµ= gµνvν

where gµν = g(eµ, eν). We call vectors, tensors, etc. all quantities that have the structure of a

vector, tensor, etc. and preserve their forms after any diffeomorphism of the manifold. We use the distinction done above to define two type of vectors.

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Definition 6.

We say that A0µ is a contravariant vector if for any f ∈ H, we have Aµ= ∂x

µ

∂x0νA 0ν

where xµ= f (x0µ).

We say that A0_µ is a covariant vector if for any f ∈ H, we have

Aµ=

∂x0ν ∂xµA

0 ν.

This definition can be extended to tensors of any dimension, joining the case of mixed com-ponents. Saying better we say that T_µν0 , T0µν, Tν0µ are respectively a covariant, a contravariant

and a mixed tensor if, with a diffeomorphism f such that xµ= f (x0µ), they transform as Tµν= ∂x0α ∂xν ∂x0β ∂xµT 0 αβ, Tµν= ∂x µ ∂x0α ∂xν ∂x0βT 0αβ_, T_νµ= ∂x µ ∂x0α ∂x0β ∂xν T 0α β .

The coordinates dxµ are an example of contravariant vector, whereas the gradient of a scalar function ∂µφ is an example of covariant vector. An example of covariant tensor is the

metric tensor as we have seen above, whereas an example of mixed tensor is the Kronecker delta δνµ, that is a sort of unity. Indeed it is possible to define the inverse of metric tensor gµν

as the contravariant tensor gνγ _{such that}

gµνgνγ = δµγ.

As we have seen above it is possible to obtain the covariant version of a vector and vice versa using the metric tensor

Aµ= gµνAν, Aµ= gµνAν.

In general, using the metric tensor it is possible to lower an upper index or to raise a lower index af any tensor. Thus in GR the line element is a general differential function of the form

ds2 = gµν(x)dxµdxν

where gµν depends on the coordinates. Mathematically gµν is the bilinear form defining the

scalar product on the tangent plane of the manifold. Thus gµν is symmetric and it has 10

inde-pendent components. Using the metric tensor it is possible to calculate some physical quantities in the space-time as time intervals and spatial distances. For example, the relationship between real time τ and the temporal coordinate x0 is given by

dτ =√g00dx0

or in integral form

τ = Z _√

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Observation 1. Clearly the expression has sense only if g00> 0. If g00< 0 in some frame of

reference means that that frame of reference could not be a physical one, i.e. it is not possible to have such a frame of reference fixed to a real body (a frame of reference is physically possible if we can fix it to a real body: we consider only these ones).

For the distances, we have dl2= −g_ij +g0ig0j g00 dxidxj = γijdxidxj (1.4)

where we have defined γij =

−gij +

g0ig0j

g00

the spatial metric tensor.

Observation 2. In general, distances does not have sense, because metric tensor depends on coordinate x0_{, so the distance from two points will have different values depending at which time}

it is measured. If metric tensor does not depend on coordinate x0, distances are well defined and are given by the integral form of (1.4).

Now we introduce some important mathematical tools from differential geometry, allowing to describe better the space-time. The first one we introduce is the covariant derivative, that extends to manifold the classical derivative of the Euclidean space.

Definition 7 (Covariant derivative).

Let M be a manifold and X (M ) the set of all vector fields on M . A covariant derivative on M is an operator

∇ : X (M ) × X (M ) −→ X (M ) (Z, Y ) 7→ ∇ZY

such that the following properties hold:

1) ∇Z(λ1Y1+ λ2Y2) = λ1∇ZY1+ λ2∇ZY2 ∀Z, Y1, Y2∈ X (M ), ∀λ1, λ2 ∈ R ;

2) ∇f1Z+f2ZY = f1∇Z1Y + f2∇Z2Y ∀Z1, Z2, Y ∈ X (M ), ∀f1, f2∈ C

∞_{(M, R) ;}

3) ∇Z(f Y ) = f ∇ZY + Y ∇Zf ∀Z, Y ∈ X (M ), ∀f ∈ C∞(M, R).

Covariant derivative allows us to define a vector field with respect to a given direction in each point of the manifold. Indeed if we take X as a tangent vector in a point p and Y as a vector field defined in a neighborhood of p, we have that the covariant derivative of the pair (v, Y ) is the derivative of the vector field Y in the direction v, and it is indicated with ∇vY .

It is important to note that in general there are infinite covariant derivatives on a manifold. If we choose a chart on the manifold, i.e. if we fix a reference frame, in term of the basis we can write that the covariant derivative of an element of the basis is

∇ρeµ= ∇eρeµ= Γ ν

µρeν (1.5)

where the symbols Γν_µρ are differentiable functions and are called the Christoffel symbols. They are not tensors for the fact that they depends on the chart we choose, but they completely describe the covariant derivative in a neighborhood of a point p. From now on we always use Christoffel symbols when we use covariant derivative, so we imply to have chosen a chart. Christoffel symbols give an operative way to compute the covariant derivative of a vector field, indeed if Yµ is a vector field, the covariant derivative of Yµ is given by

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and for the covariant version Yµwe have

∇_ρYµ= ∂ρYµ+ ΓνρµYν. (1.6)

The definition can be extended to a general tensor field Tµ1...µp ν1...νq ∇_ρTµ1...µp ν1...νq = ∂ρT µ1...µp ν1...νq + p X i=1 Γµi ρδT µ1...µi−1δµi+1...µp ν1...νq − q X i=1 Γδ_ρν_iTµ1...µp ν1...νi−1δνi+1...νq. (1.7)

Together with covariant derivative we can introduce the covariant differential of a contravariant vector Aµ as

DYµ= dxα∇αYµ= dxα(∂αYµ+ ΓµανYν) , (1.8)

and the definition can be extended to covariant vectors using (1.6), and in general to all types of tensors using (1.7). The concept of covariant derivative is essentially equivalent to the connection, i.e. a mathematical object that allow to connect tangent spaces in different points of the manifold. So we use connection to identify Christoffel symbols. From connection is possible to define a tensor, the torsion tensor that is the antisymmetric part of the connection

S_µνλ = 1 2(Γ

λ

µν− Γλνµ) (1.9)

where we have introduced a new notation, writing upper index after the lowers. Until now we have always written tensor with lower and upper indexes at the same level, suggesting that when we lower an upper index we use the rule

gλγSµνγ = Sλµν

With the notation S_µνλ we mean that in order to lower an upper index we use gλγSµνγ= Sµνλ.

It is possible to demonstrate that S_µνλ is a tensor. Now we introduce a special class of real smooth metric manifolds.

Definition 8. A Riemannian manifold is a pair (M, g), where M is a real smooth manifold and g is a symmetric bilinear form of the tangent space on M , of signature (n,0), where n is the dimension of M , meaning that the bilinear form g is positive-definite.

A pseudo-Riemannian manifold is a pair (N, h), where N is a real smooth manifold and h is a symmetric bilinear form of the tangent space on N , that is non-degenerate.

In particular space-time of GR is a pseudo-Riemannian manifold.

In the case of Riemannian and pseudo-Riemannian manifolds, there is a unique connection that satisfies these two properties:

• ∇_ρgµν= 0, i.e. the connection is compatible with the metric,

• Γρµν = Γρνµ, i.e. the connection is symmetric.

This connection is called the Levi-Civita connection and in coordinates it is defined in terms of the metric tensor as

Γα_µν = 1 2g

λα_(∂

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This is the connection that Einstein used to construct the space-time of GR. This choise was made because the two properties satisfied by Levi-Civita connection seemed to be reasonable for space-time. Clearly if we choose the Levi-Civita connection from (1.9) we have that for the torsion tensor holds

S_µνλ = 1 2

Γλ_µν− Γλ_νµ= 0

and we say that the manifold is torsion-free. Thus space-time in GR is torsion-free, but we will see later some alternative theories of gravity admitting a nonvanishing torsion tensor.

We now introduce another instrument of differential geometry, typical of Riemannian man-ifold, that can also be defined for any pseudo-Riemannian manifold. It is an instrument that allows to study the intrinsic curvature of a differential manifold. This object is a tensor, called Riemann tensor and it is defined in terms of Christoffel symbols as

Rµ_αβγ = ∂βΓµαγ− ∂γΓµ_αβ+ Γµ_βνΓναγ− ΓµγνΓναβ. (1.11)

As we said, Riemann tensor is related to the curvature of the manifold, indeed it is possible to demonstrate that a manifold is flat if and only if Rµ_αβγ = 0 everywhere. Thus Riemannian tensor allows us to distinguish if a manifold is flat or curved. In the case of GR, this means that we can distinguish if there is a gravitational field or not.

Observation 3. Although it is always possible to choose a locally flat set of coordinates, i.e. gµν(xλ) = ηµν in some point xλ and so Γµ_αβ(xλ) = 0, Riemann tensor does not vanish in that

point, because in general ∂γΓµ_αβ(xλ) 6= 0, so in this case its expression becomes

Rµ_αβγ(xλ) = ∂βΓµαγ(xλ) − ∂γΓµ_αβ(xλ).

The Schwartz theorem on second derivatives does not apply to covariant derivatives. We can compute the difference between mixed second covariant derivatives for a covariant vector Aα using the Riemann tensor by the rule

∇µ∇νAα− ∇ν∇µAα = −AλRαµνλ .

A similar expression holds for contravariant vector. We now enunciate some properties of the Riemann tensor that will bee useful in the next sections.

• Symmetry proprieties

Rαβγδ= −Rβαγδ= −Rαβδγ= Rβαδγ

Rαβγδ= Rγδαβ

• First Bianchi identity

Rαβγδ+ Rαγδβ+ Rαδβγ= 0;

• Second Bianchi identity

∇_δRµ_αβγ + ∇γRµ_αδβ+ ∇βRµ_αγδ = 0.

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1.2. EINSTEIN FIELD EQUATIONS 15

• the Ricci tensor that is defined as

Rαβ = Rανβν = gµνRµανβ; (1.12)

• the scalar curvature that is defined as

R = Rβ_β = gαβRαβ = gαβgµνRµανβ. (1.13)

Riemann tensor also appears in the equation of relative motion of two bodies by D2ξµ

ds2 = ξ

ν_uα_uβ_Rµ βαν

where ξµ= xµ₂ − xµ₁ is relative distance and uµ is relative velocity.

1.2 Einstein field equations

There are many ways to determine the equations describing the motion of the matter in a gravitational field. Here we use the variational approach. Let us consider the action functional for a gravitational field: it will be of the form

Sg =

Z Lg

√

−gdΩ (1.14)

where Lg is the Lagrangian density of the gravitational field, g is the determinant of metric

tensor gµν and dΩ = dx0dx1dx2dx3 is the infinitesimal quadrivolume element. The Lagrangian

density must be a scalar and can not contain second order or higher derivatives of the metric tensor. The only scalar quantity we need is the scalar curvature R, but it contains the second order derivatives of the metric tensor. It is possible to prove that only first derivatives contribute to the integral so Lg = −_16πG1 R could be a good choice. In order to obtain the variation of

(1.14), we compute the variation of the scalar curvature R

δR = δ(gµνRµν) = gµνδRµν+ Rµνδgµν and also δ√−g = − δg 2√−g = ggµνδgµν 2√−g = − 1 2 √ −gg_µνδgµν. Thus for the action we have

δSg= − 1 16πG Z gµνδRµν+ Rµνδgµν− 1 2Rgµνδg µν _√ −gdΩ. Integrating by parts the first term we obtain

δSg = − 1 16πG Z Rµν− 1 2Rgµν δgµν√−gdΩ. The total action will be of the form

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where Sm is the action of matter. The variation of Sm is δSm= 1 2 Z Tµνδgµν √ −gdΩ

where Tµν is the stress-energy momentum tensor. The stress-energy momentum tensor is a

tensor quantity that describes the density and flux of energy and momentum in space-time. It is an attribute of matter, radiation, and non-gravitational force fields. In SR, Tµν fulfills the

following equations

∂µTµν = 0 (1.15)

expressing the conservation of nongravitational energy, linear and angular momentum. For the principle of general covariance, in GR equations (1.15) become

∇µTµν = 0. (1.16)

In this case, equations (1.16) do not represent any conservation law. It is possible to find also in GR a quantity that behave as the stress-energy momentum tensor in SR. This quantity is the stress-energy momentum pseudo-tensor. Pseudo-tensor because is not a tensor. In this work we do not focus on stress-energy momentum tensor. For more details see [Landau-Lifsits]. So the total variation of the action functional is

δS = Z − 1 16πG Rµν − 1 2Rgµν +1 2Tµν δgµν√−gdΩ. Imposing δS = 0, we obtain the so called Einstein field equations

Rµν−

1

2Rgµν = 8πGTµν. (1.17)

This is a group of 10 independent nonlinear partial differential equations. The variables of the equations are the components of metric tensor gµν, but also the equations for the matter

are included in (1.17). Thus the dynamics of the matter is determined solving the Einstein equations.

Schwarzschild’s solution

The first non trivial solution of the Einstein equations was the one by Karl Schwarzschild, found in 1915 and published in 1916. It expresses the gravitational field generated by a massive non-rotating body. In order to found this solution we have to consider a gravitational field with central symmetry. It means that the line element ds has the same expression in points having the same distance from the center of the field. Using spatial spherical coordinates r0, θ and ϕ, the general expression for ds2 is

ds2 = g00(r0, t0)dt02+ g01(r0, t0)dt0dr0+ g11(r0, t0)dr02+ Φ(r0, t0)dΩ2

where dΩ2 = dθ2+ sin2θdϕ2. In GR the choice of the reference frame is arbitrary, so we can make a change of the coordinates (r0, t0) ← (r, t) such that

g01(r, t) = 0 Φ(r, t) = −r2.

So for the line element we have

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1.2. EINSTEIN FIELD EQUATIONS 17

that we can rewrite as

ds2 = eν(r,t)dt2− eλ(r,t)dr2− r2dΩ2 (1.18) using the exponential notation. We also suppose to be enough distant from the mass that generated the field, so we can consider that there is no matter and Tµν = 0. From the expression of the metric tensor (1.18) we can find the Levi-Civita connection Γα_µν. Using the notations ˙ = ∂/∂t and0 = ∂/∂r, we have that the nonvanishing Christoffel symbols are

Γ0₀₀= ν˙ 2, Γ 0 01= ν0 2, Γ 0 11= 1 2e λ−ν_˙λ, _Γ1 00= 1 2e ν−λ_ν0 Γ1₀₁= 1 2˙λ, Γ 1 11= 1 2λ 0_, _Γ1

22= −re−λ, Γ133= −re−λsin2θ,

Γ2₁₂= Γ3₁₃= 1 r, Γ 2 33= − sin θ cos θ, Γ323= cos θ sin θ.

Now using relationship (1.11) and the definition of Ricci tensor (1.12) we find that the only component of Rµν with different indexes is R01 = ˙λ/r. Thus from (1.17) we find that ˙λ = 0,

that is λ = λ(r). The Einstein field equations reduce to ν” +1 2ν 0 (ν0− λ0) +2ν 0 r = 0, (1.19) ν” +1 2ν 0_(ν0_{− λ}0_{) −}2λ0 r = 0, (1.20) r 2(λ 0_{− ν}0 )e−λ+ 1 − e−λ= 0, (1.21) hr 2(λ 0_{− ν}0 )e−λ+ 1 − e−λ i sin2θ = 0. (1.22)

From the first two equations we find

λ0+ ν0 = 0 ⇒ ν(t, r) = −λ(r) + f (t) (1.23) with f (t) an arbitrary function of coordinate t. Substituting (1.23) in (1.21) we have

rλ0e−λ+ 1 − e−λ = 0 ⇒ e −λ_λ0 e−λ_{− 1} = 1 r ⇒ − log(|e−λ− 1|) = log r + k ⇒ e−λ = 1 −h r with h a constant to be determined. Again from (1.23) we obtain

eν = H(t)

1 −h r

with H(t) > 0 a function of time. It is possible to eliminate the function H(t) with a change of coordinate for the time t given by t0 =R pH(t)dt, so we can suppose H(t) = 1. Thus the expression of the line element is

ds2= 1 −h r dt2− 1 1 −h_rdr 2_{− r}2_dΩ2 _(1.24)

In order to establish the value of the constant h, we impose that at great distances g00 has the

same expression of the non-relativistic case, i.e. the Newtonian case, which is ˜

g00= 1 + 2ϕ = 1 −

2M r

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where M is the gravitational mass generating the field. Thus we find that h = 2M and we call rg = h = 2M the gravitational radius or the Schwarzschild radius.Using non-geometric units

the last expression becomes

rg = 2GM c2 So we can rewrite (1.24) as ds2 =1 −rg r dt2− 1 1 −rg r dr2− r2_dΩ2 _(1.25)

that is known as Schwarzschild metric. The metric that defines (1.25) represents the gravita-tional field generated by a massive body as a modification of the space-time.

Now we give a look to some properties of the Schwarzschild metric, for a more detailed explanation see [Landau-Lifsits]. The first property we express is that Schwarzschild metric is stationary, i.e. g0i= 0, so the spatial metric tensor is simply

γij = −gij +

g0ig0j

g00

= −gij

and the spatial element line is

dl2= 1 1 −rg

r

dr2+ r2dΩ2.

Thus distances are well defined because the metric tensor is not time dependent. This property can be expressed saying that the metric is static. The distance between two point r1, r2 lying

on the same radius, i.e. with dΩ = 0 is given by the integral Z r1 r2 dr q 1 −rg r > r2− r1

The geometrical meaning of the coordinate r is that the length of the circumference of radius given by the coordinate r is 2πr. For physical time we have the expression

dτ =√g00dt =

r 1 −rg

r dt ≤ dt

so time gets slower near masses. This fact causes the gravitational redshift as we will see later. It is also possible to synchronize the clock because the metric is stationary, i.e. g0i = 0 for all

i. Moreover the Schwarzschild metric tensor becomes singular for r = 0 and r = rg. But these

two singularities have different nature: r = 0 is a singularity due to space-time and it is not avoidable, whereas r = rg is due to coordinates. Indeed it is possible to make a suitable change

of coordinates that eliminates the singularity in r = rg, for example passing to hyperbolic

coordinates.

1.3 Equations of motion in GR

One way to derive the equation of motion of a body in a curved space is using a variational principle. The action functional for a test body of mass m in curved space-time is

S = −m Z

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1.3. EQUATIONS OF MOTION IN GR 19

where ds2 = gµνdxµdxν. Computing the first variation of (1.26) and imposing it equal to zero,

we obtain δS = −m Z δ(ds) = −m Z ₁ 2ds[2gµνdx µ_δ(dxν_{) + δg} µνdxµdxν] = −m Z [gµαuµd(δxν) + ∂αgµνuµdxνδxα] = −m Z uα d dsδx α₊1 2∂αgµνu µdxν ds ds = m Z _d2_x α ds2 − 1 2∂αgµν dxµ ds dxν ds δxαds = 0

where we have introduced the four-velocity uµ = dx

µ

ds . Applying the variational principle we obtain d2xα ds2 − 1 2∂αgµν dxµ ds dxν ds = 0 .

Using the expression uµ = gµνuν and the definition of Levi-Civita connection (1.10), this

expression can be written as

Duα ds = duα ds − Γ β ανuβuν = 0

or in term of covariant coordinates xµ

d2xα ds2 − Γ µ αν dxµ ds dxν ds = 0. (1.27)

For contravariant version we have d2xα ds2 + Γ µ αν dxµ ds dxν ds = 0 (1.28)

where we have used

Duµ ds = g

µαDuα

ds = 0 (1.29)

because from the definition of Levi-Civita connection we have Dgµν = 0.

These are the equations of motion of a body in a curved space, the geodesics of the metric gµν that we can compute in any smooth metric manifold. These equations are the covariant

form of the equations d2xµ/ds2 = 0 that holds in SR. Now we study more in details the motion of a body in a space-time equipped with Schwarzschild metric (1.25). Let us consider a particle of mass m in a gravitational field. Since we are in presence of a central field, the motion must occur in a single plane passing through the center of the field. Thus, changing the reference frame, we can suppose that the plane of the motion is the one expressed by θ = π₂. The expression of the line element becomes

ds2 = 1 −rg r dt2− 1 1 −rg r dr2− r2dϕ2. Let us consider the equivalent form of equations (1.27) given by

dpα

ds = 1

2m∂αgµνp

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Since t and ϕ are cyclic variables, the quantities p0 and p3 are constants of motion, so we can

fix them as

p0 = mE p3 = −mL

where E and L are the energy and the angular momentum of the particle, computed with its initial conditions. Now we have

p0 = mdt

ds, (1.30)

and using the relation

p0 = gµ0pµ= g00p0= 1 −rg r −1 mE we obtain the equation for coordinate t, that is

dt ds = 1 −rg r −1 E. (1.31)

In the same way for p3 we have

p3 = mdϕ

ds. (1.32)

Using the relation

p3 = gµ3pµ= g33p3=

mL r2

we find the equation for the coordinate ϕ dϕ

ds = L

r2. (1.33)

In order to find the equation for the coordinate r we have to use the relation pµpµ= m2. From

the other hand we have

pµpµ= p0p0+ p1p1+ p2p2+ p3p3= g00(p0)2+ g11(p1)2+ g33(p3)2

and substituting expressions (1.25),(1.30) and (1.32) we obtain pµpµ= 1 −rg r −1 m2E2−1 −rg r −1 m2 dr ds 2 −m 2_L2 r2 = m 2_.

Resolving it for dr_ds we finally have dr ds 2 = E2−1 −rg r 1 +L 2 r2 . (1.34)

The equations (1.34),(1.33) and (1.31) represent the equations of motion of a particle in a gravitational field with central symmetry. Working on the equation (1.34), we can see that there are four possible type of motion:

• circular (stable and unstable); • elliptic;

• hyperbolic;

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1.3. EQUATIONS OF MOTION IN GR 21

The unstable circular orbit and the motion that predict that the particle falls in the singularity are not present in Newton’s theory. In a similar way we can find the equations of motion for a light ray. We know that light rays fulfill the equation ds = 0 and also the equations for electromagnetic waves ∂µ∂νAµ= 0. The form of an electromagnetic wave is

Aµ= εµeikνx ν

where kµ = dx

µ

dλ , with λ a parameter. In SR we have the equations dk

µ _{= 0 and k}µ_k µ = 0.

Using the general covariance principle we have that in GR kµ _{fulfills the equations}

kµkµ= 0, Dkµ= 0 ⇒ dkµ dλ = 1 2∂µgαβk α_kβ_.

As for the equations of motion for massive body, we find that the quantities k0 and k3 are

constants during the motion so we can put

k0= E, k3= −L.

We first find the equation of motion for the t component. Matching the two equations k0= dt dλ, k 0_{= g}00_k 0 = 1 −rg r −1 E we obtain dt dλ = 1 −rg r −1 E. (1.35)

Similarly for the ϕ component we have k3 = dϕ dλ, k 3_{= g}3µ_k µ= g33k3= L r2

and mixing the equations we obtain

dϕ dλ =

L

r2. (1.36)

Finally in order to find the equation for r component we have to use 0 = kµkµ= g00(k0)2+ g11(k1)2+ g33(k3)2 joined with k1= dr dλ. So we obtain 1 −rg r −1 E2−1 −rg r −1 dr dλ 2 −L 2 r2 = 0.

From this equation we find

dr dλ 2 = E2−1 −rg r L2 r2 (1.37)

The equations (1.37), (1.36) and (1.35) are the equations of motion of a light ray in the Schwarzschild space-time.

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1.4 Four classical tests of GR

Just from the beginning GR succeeded in explain some phenomena still unsolved for classical physics. These phenomena, once verified with experiments, are known as the four classical tests of GR. Now we give a look at these ones and we see how they can be explained with GR.

Precession of perihelion

In the case of elliptical motion, like planet’s one, GR predicts the phenomenon of the precession of the perihelion. Using equations (1.34) and (1.33) we can write

dr dϕ = r2 L s E2₋_{1 −}rg r 1 +L 2 r2

Now separating the variables dr and dϕ we obtain dϕ = L r2 dr r E2_{− 1 −} rg r 1 +L_r22 (1.38)

Integrating (1.38) between rmin and rmax we should obtain π: but the integral is in general

different from π. The difference is given by

∆ϕ = 2     Z rmax rmin L r2 dr r E2_{− 1 −} rg r 1 +L_r22 − π     (1.39)

In order to find rmin, rmax, we have to solve the equation

E2−1 −rg r 1 +L 2 r2 = 0 (1.40)

In order to compute (1.39), is useful making the change of coordinates u = 1/r, so we have umax = 1/rmin and umin = 1/rmax, and the equation (1.40) becomes

E2− (1 − rgu)(1 + L2u2) = 0

The last expression is a third degree polynomial with two zeros given by umin and umax, so we

can rewrite the left hand side like

−L2(u − umax)(u − umin)[1 − rg(u + umax+ umin)]

Replacing this expression in the integral (1.39) we obtain ∆ϕ = 2

" Z umax

umin

du

p−(u − umax)(u − umin)[1 − rg(u + umax+ umin)]

− π #

We can assume that the quantity rg(u+umax+umin) is much smaller than 1 (the Schwarzschild

radius of the body that generates the field is much smaller than the distance of the body that fill the precession), so we can develop in Taylor series the denominator obtaining

∆ϕ ' 2 (

Z umax umin

[1 +rg

2(u + umax+ umin)]du

p−(u − umax)(u − umin)

− π )

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1.4. FOUR CLASSICAL TESTS OF GR 23

that we can write as ∆ϕ ' 2 h 1 +rg 2(umax+ umin) iZ umax umin du

+ rg

Z umax umin

u du

− 2π We can easily compute the two integrals with the substitution

p−(u − umax)(u − umin) = (u − umin)x

obtaining the expression ∆ϕ ' 2π h 1 +rg 2 (umax+ umin) + rg 4(umax+ umin) − 1 i = 3 2πrg(umax+ umin) Writing this expression in terms of rmax and rmin we have

∆ϕ ' 3 2πrg

rmin+ rmax

rminrmax

Using Mercury’s values for those quantities we obtain

∆ϕ ' 5.09 × 10−7rad/turn = 43.2”/100 yr

in agreement with the experimental value, that was not consistent with the Newtonian estimate.

Deflection of the light

Differently from classical Newtonian theory, GR predicts deflection of the light due to gravita-tional field. Let us consider a light beam coming from infinity with a given impact parameter. Mixing the equations (1.34) and (1.33) we have

dr dϕ = r2 L s E2₋ 1 −rg r L2 r2 = r2 s E L 2 − 1 r2 1 −rg r . Now separating the variables we obtain the equation

dϕ = dr r2 s E L 2 − 1 r2 1 −rg r (1.41)

Integrating this quantity between ∞ and rmin, where rmin is the minimal distance between

the path of light and the symmetry center (that can be found resolving the equation E_L22 −

1 −rg r

₁

r2 = 0), if there is no deflection of light, assuming the path parallel to the x-axis, the

coordinate ϕ should vary of π/2. But it is easy to check that the integral in (1.41) is not π/2. Using elementary geometrical arguments, we have that the discrepancy between the integral and π/2 is a half of the deflection angle, so we have

∆ϕ = −2   Z ∞ rmin dr r2 q E L 2 − 1 r2 1 − rg r +π 2  . (1.42)

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Figure 1.1: The deflection of a light ray passing near a massive body.

As before we can estimate (1.42) making the transformation given by u = 1/r and using Taylor expansion near small quantities, obtaining the final expression

∆ϕ ' 2 rg rmin

that measure the angle of deflection of a light path passing near a massive body (Fig.1.1). It is possible to check that this quantity is exactly the double of the quantity that can be found with classical Newton physics. The accuracy of Einstein’s formula has been confirmed by experiments.

Gravitational red shift

Another phenomenon predicted by GR and in agreement with observations is the gravitational red shift. One way to show this effect is the following: let us consider a stationary metric tensor gµν, like the Schwarzschild’s one, and a light signal starting from a point A with spatial

coordinates xi+ dxi, arriving to the point B with spatial coordinates xi, being reflected and coming back to the point A as in Fig.1.2. The equation fulfilled by the light signal is clearly ds = 0 where

ds2= gµνdxµdxν = g00(dx0)2+ 2g0idx0dxi+ gijdxidxj

Imposing ds = 0, we find two solution for dx0 dx0_1,2= 1 g00 −g_0idxi∓ q (g0ig0j − gijg00)dxidxj (1.43)

If x0 is the coordinate time for the point B(when the light arrives in B), we have that coordinate time of signal in point A is x0 + dx0₁ when it starts and x0+ dx0₂ when it comes back. We now define as simultaneous events the points

(x0, xi), (x0+dx

0 1+ dx02

2 , x

i_{+ dx}i₎

Calling δx0 the difference between two simultaneous infinitesimal events we have δx0 = dx 0 1+ dx02 2 = − g0i g00 dxi (1.44)

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Figure 1.2: The light ray starting from the point A of spatial coordinates xi+ dxi, arriving at the point B with spatial coordinates xi and coming back to A.

This expression allows us to synchronize clocks allover in a not-closed path. In the case of closed paths it is possible to synchronize clocks only if we have a static metric tensor, i.e. if g0i= 0 for all i. Let us now consider two simultaneous events

(x0₁, xi) (x0₂, xi+ dxi) We know from equation (1.44) that coordinates time fulfill

x0₂− x0₁= −g0i(x

0 2, xi)

g00(x02, xi)

dxi. (1.45)

If we now increase coordinate time of a same quantity δx0, using the same formula we have x0₂+ δx0− x0₁− δx0 = −g0i(x

0

2+ δx0, xi)

g00(x02+ δx0, xi)

dxi that is the same of (1.45) if the metric tensor is stationary, so the events

(x0₁+ δx0, xi) (x0₂+ δx0, xi+ dxi)

are still simultaneous. What is different in general is the physical time occurred in the spatial points, indeed we have

δτ1 = q g00(xi)δx0 δτ2 = q g00(xi+ dxi)δx0

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This is the cause of gravitational red shift. In order to explain this effect let us consider two light signal starting from a source with spatial coordinates xi, with coordinates time x0₁ and x0₂ (we can imagine that are two peaks of an electromagnetic wave). The signals arrive to an observer in some place of the space respectively in ∆x0₁ and ∆x0₂. The quantities ∆x0₁ and ∆x0₂ can be computed using

∆x0∗ =

Z

dx0∗ (1.46)

where dx0∗are solutions of the equation ds2= 0 for the two signals, which expression is given by

(1.43), and ∗ can assume value 1 or 2. Every terms of (1.46) must be computed in coordinates of light signal, so we have

∆x0∗ = Z dx0∗ = Z −g0i g00 (x0∗, xi)dxi± 1 √ g00 (x0∗, xi)dl where dl2 _{= γ}

ijdxidxj and γij is the spatial metric tensor. We can observe that if we have a

stationary metric tensor, we are going to have ∆x0₁ = ∆x0₂, so the two signal employ the same time to arrive to the observer. So if we call δx0 the coordinate time gap between the start of the signals, we have the same gap at the arrive. But in general the physical time passing between the signals at the start and at the arrive are different, indeed we have that the physical times are δτ1 = p g00(S)δx0 δτ2 = p g00(O)δx0

where S and O are the coordinates of the source and of the observer respectively. So the relation between physical times is

δτ1

pg00(S)

= δτ1 pg00(O)

. (1.47)

If we imagine that the two signals represent two consecutive peaks of an electromagnetic wave, we can compute the wave frequency at the start and at the arrive and we obtain

ωS = 2π δτ1 , ωO = 2π δτ2

and using (1.47) we have the relation

ωO= ωS

s g00(S)

g00(O)

. (1.48)

In case of weak field, we can approximate g00 with its Taylor expansion

g00(S) = 1 + 2ϕS, g00(O) = 1 + 2ϕO (1.49)

where ϕSand ϕOare the gravitational potential at the source and at the observer. Substituting

(1.49) in (1.48) we obtain

ωO = ωS

r 1 + 2ϕS

1 + 2ϕO

' (1 + ϕS)(1 − ϕO)ωS

and at the first order in the potentials we have ωO− ωS

ωS

= ϕS− ϕO.

If |ϕO| < |ϕS| we have ωO < ωS, i.e. there is a shift towards low frequencies. For this reason

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Light-time delay

The last classical test of GR is the light-time delay also known as Shapiro delay in honor of Irwin Ira Shapiro, an American astrophysicist that discovered this effect in 1964 bouncing radar beams off the surface of Venus and Mercury, and measuring the round trip travel time. This effect is more recent with respect to the previous three tests, but it is often included into the classical tests of GR. The light-time delay measured by Shapiro is caused by the fact that in GR the speed of light is slower when the light passes near a massive body.

In the Schwarzschild space-time, where the gravitational field is generated by the Sun we consider two points, for example the Earth and a planet, with r1, r2 their respectively radial

coordinates, and a light signal starting from the Earth, arriving to the planet and being reflected to the Earth.

The signal fulfills the equations for light signal that we have found below. Thus using equation (1.35) we obtain dr dλ = dr dt dt dλ = ˙r 1 −rg r (1.50)

where we have fixed E = 1. Now, substituting in the equation of the light ray ds = 0, the expression of the metric tensor and the expressions (1.50) and (1.33) we obtain

0 = 1 1 −rg r − ˙r 2 1 −rg r 3 − L2 r2.

Resolving for dt/dr we find dt dr = 1 1 −rg r " 1 −1 − rg r 1 −rg r0 r₀ r 2 #−1/2 (1.51) where r0is the solution of dr/dλ = 0, i.e. the point of minimal approach to the Sun. Developing

in Taylor series the term in the square bracket we have 1 −1 − rg r 1 −rg r0 r₀ r 2 ' 1 − 1 −rg r + rg r0 r₀ r 2 ' 1 −r0 r 2 1 − rgr0 r(r + r0) . Thus the expression (1.51) can be rewritten as

dt dr ' 1 1 −rg r 1 −r0 r 2−1/2 1 − rgr0 r(r + r0) −1/2 ' _q 1 1 − r0 r 2 1 +1 2 rgr0 r(r + r0) +rg r .

Now separating the variable and integrating between r1 and r0 we obtain

∆tr1r0 = Z r1 r0 1 q 1 − r0 r 2 1 +1 2 rgr0 r(r + r0) +rg r dr = q r2 1− r02+ rg ln r 2 1− r02 r0 +1 2 q r2 1 − r02 (1.52) where the first term is the Newtonian light-time, whereas the second term is the relativistic correction. In order to compute the total light-time we have to apply the same computations for the path between r0 and r2. Thus the total light time is given by

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This is just one possible formulation of the light-time delay. As we will see in Chapter 3, a more accurate formulation for the light-time is needed by reason of better sensibility of the instruments used during spatial missions.

During years many other tests have confirmed the validity of GR. Between these tests we mention gravitational lensing, frame-dragging tests, existence of black holes, binary pulsars, detection of gravitational waves and expansion of the universe.

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Chapter 2

Alternative metric theories of

gravity

In this chapter we are going to introduce alternative metric theories of gravity. In the first part we will concentrate on the Parametrized Post-Newtonian (PPN) formalism, a tool that express, using some parameters, the field equations in the particular limit of weak-fields and slow-motion. In the second part we will introduce a class of metric theories of gravity, admitting a torsion tensor different from zero: after the necessary mathematical background, we will discuss about the physical purpose beyond this concept. After that, following [March et al. 2011] and [Mao et al. 2007], we will update the PPN formalism with new parameters for investigating the torsion, and we will compute the equations of motion of a body in this new formalism.

2.1 Post-Newtonian formalism

After the birth of GR many others theories of gravity were developed. In this section we focus on an important class of alternative theories of gravity: the metric theories. These theories, like GR, admit a space-time governed by a metric tensor and the gravitation is seen as a geometrical phenomenon. An important principle to distinguish between metric theories of gravity and non-metric ones is the Einstein Equivalence Principle (EEP).

Definition 9 (Einstein Equivalence Principle). The following statements holds true: (i) WEP is valid;

(ii) the outcome of any local nongravitational test experiment is independent of the velocity of the freely falling apparatus;

(iii) the outcome of any local nongravitational test experiment is independent of where and when in the universe it is performed.

This definition permits to divide gravitational theories in two distinct classes: metric the-ories, those that embody EEP, and nonmetric thethe-ories, those that do not embody EEP. The reason is the following: if EEP is valid, gravitation must satisfy the three postulates of a metric theory of gravity:

(i) space-time is endowed with a metric gµν;

(ii) the world line of test bodies are geodesics of the metric; 29

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(iii) in local freely falling frames the nongravitational laws of physics are those of special relativity.

A consequence of EEP is that matter and nongravitational fields are completely oblivious to them. Metric is the only quantity that enters the equation of motions, the other fields may help to generate the space-curvature associated with the metric, but they cannot interact directly with the matter: matter responds only to metric. This concept was well expressed by John Archibald Wheeler

”Space-time tells matter how to move; matter tells space-time how to curve. ”

So matter and metric are the primary theoretical entities in every metric theory, and they distinguished from each other for the particular way in which matter generates the metric. In the class of metric theory is possible to make a further division into two classes: purely dynamical theories and prior-geometrical ones. The distinction can be explained as follows.

• A purely dynamical theory is a metric theory whose gravitational fields are determined by coupled partial differential fields equations, maybe with the addiction of some other field. In this context it is possible to distinguish between scalar theories, vector theories and tensor theories, depending on whether external field appears in the theory in addiction to metric tensor. These fields are usually indicate with φ, Kµ, Bµν respectively.

• A prior-geometrical theory is a metric theory that admit absolute elements, that are fields or equations that are given a priori and are independent of the structure and evolution of other fields. These absolute elements could be a flat background metric ηµν or cosmic

absolute time T as examples.

Clearly not all the possible metric theories could be taken into account. We must consider only those theories satisfying at least three criteria for viability:

• self-consistency; • completeness;

• agreement with past experiments.

The meaning of the first property, the self-consistency, is that predictions of a theory must be unique, that is if we calculate the predictions of some experiments using two different but equivalent methods, we always gets the same results. The second property, the completeness, means that a candidate theory must be capable of analyzing the outcome of any experiment of interest: the theory must incorporate a complete set of electrodynamic and quantum me-chanical laws. Clearly we can not ask the completeness in such areas like weak and strong interaction, quantum gravity, unified field theories because even special and general relativity are not considered complete or fully developed, so what we want is that a viable theory contains a set of gravitationally modified Maxwell equations. The third property, the agreement with past experiments, means that are considered only those theories which predict and explain the results of past experiments, included experiments involving special relativity. Similarly a viable gravitation theory must have as limit for gravity turned off respect to other interactions, the nongravitational laws of special relativity and also as limit for weak fields and slow motion the Newton’s gravitational laws.

In order to verify a certain metric theory, we need to compare its predictions with exper-iments, most of which could be performed in Solar System, that is a region of weak-field and

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2.1. POST-NEWTONIAN FORMALISM 31

slow-motion. Thus, in order to compare theories with Solar System experiments we could use just the weak-field and slow-motion limit of such theories. This limit is called Post-Newtonian limit. In this context a general theory has been developed during the second half of 20th cen-tury: the Parametrized Post-Newtonian (PPN) formalism. In next section we are going to describe PPN formalism and we will show how to calculate the PPN expansion for a generic metric theory of gravity. In this formalism we could compare different metric theories in an easy way.

2.1.1 Parameterized Post-Newtonian formalism

We start by the fact that Newton’s gravitational law adequately explain experiments to an accuracy of about one part in 105, so Newtonian physics may be viewed as a first-order ap-proximation of a metric theory of gravity. The idea is to develop the quantities that appear in metric theory in series of small parameters, with first-order approximation given by Newtonian physics laws. In the Solar System there are many quantities that can considered small:

• Newtonian gravitational potential U is nowhere larger than 10−5_;

• planetary velocities that are related to U by v2_{. U ;}

• the quantity p/ρ, where p is the pressure of the matter and ρ is the mass density, that is related to U by p/ρ . U ;

• the specific energy density Π that is related to U by Π . U. At these quantities we can assign an order of smallness, saying

U ∼ v2∼ p/ρ ∼ Π ∼ O(2).

The quantities p, ρ, Π appear in the stress-energy momentum tensor Tµν. In this work we do not treat the mathematical and physical formulation of the stress-energy momentum tensor, we will consider it known, for more details see [Will 1993]. Now we built a general Post-Newtonian metric theory and in the next Section we will see how to calculate Post-Newtonian limit for any metric theory. We start using Newtonian physics: for a test body we have

~a = ∇U (2.1)

where ~a is the acceleration of the body and U is the Newtonian potential produced by rest-mass density ρ according to

∇2U = −4πρ, U (~x, t) =

Z _ρ(~_x0_{, t)} |~x − ~x0_|d

3_x0_.

Using geodetics equations (1.28), holding in every metric theory, in a static coordinates system for the body’s acceleration we have

d2xk dt2 = a k _{= −Γ}k 00= 1 2g kl_g 00,l (2.2)

where g00,l = ∂g_dx00l . In presence of a very weak gravitational field, equations (2.1) and (2.2) give

the same expression only if

gjk ' δjk_, _g

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So for any metric theory, first-order approximation of its metric tensor must have the form (2.3). In order to describe the general Post-Newtonian metric tensor, we use the action for a single particle that for any metric theory is

S = −m0 Z −gµν dxµ dt dxν dt 1₂ dt = −m0 Z (−g00− 2g0jvj− gijvivj) 1 2dt

where v is the velocity of the particle. The function in the integral can be seen as a Lagrangian L of a single particle. We observe that Newtonian limit corresponds to

L = (1 − 2U − v2)12

so, using the order of smallness introduced before, we see that Newtonian physics is given by an approximation for L correct to O(2). In order to construct a Post-Newtonian theory we need therefore at least an approximation of next highest order, O(4). Terms of order O(1) and O(3) can’t appear in the Lagrangian L for the fact that conservation of the rest mass and conservation of energy would be violated. Therefore we require the knowledge of

g00 to O(4),

g0j to O(3),

gij to O(2).

(2.4)

Now we give a brief summary of the steps bringing to the final form of Post-Newtonian metric tensor, for details see [Will 1993]. The first step is to specify the coordinate system. We imagine an homogeneous isotropic universe and an isolated Post-Newtonian system. The idea is to use a reference frame were the outers region are in free fall with respect to the rest frame, the reference frame where the universe is at rest, and where the metric in the region of the system is given by

ds2 = (ηµν+ hµν)dxµdxν

where hµν is the deviation from Minkowskian metric tensor due to the system. This type

of choice is possible, as we can see in [Will 1993]. Now we introduce some quantities, the potentials, used to construct the general form of metric tensor, taking into account (2.4). For this point we also need to follow some rules arising from experiences and evidences:

• metric tensor should contain only Newtonian or Post-Newtonian order terms, no higher terms are included;

• terms should tend to zero as the distance becomes large; • metric should be dimensionless;

• spatial origin and initial moment of time are completely arbitrary, so the metric should contain no explicit reference to these quantities;

• h₀₀, h0j, hij should transform as scalar, vector, tensor under spatial rotations;

• potentials should be generated by rest mass, energy, pressure and velocity, not by their gradients;

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Following these rules and taking in account (2.4) the only terms that can appear are U = Z _ρ0 |~x − ~x0|d 3_x0 Uij Z _ρ0_{(x − x}0₎ i(x − x0)j |~x − ~x0|3 d 3_x0 Vj = Z _ρ(~_x0_{, t)v}0 j |~x − ~x0_| d 3_x0 Wj = Z _ρ(~_x0_{, t)~}_v0_{· (~}_{x − ~}_x0_{)(x − x}0₎ j |~x − ~x0|3 d 3_x0 ΦW = Z ρ0ρ00 ~x − ~x 0 |~x − ~x|3 · ~x0− ~x00 |~x − ~x00_|− ~ x − ~x00 |~x0_{− ~}_x00_| d3x0d3x00 Φ1= Z _ρ0_v02 |~x − ~x0|d 3_x0 Φ2= Z ρ0U0 |~x − ~x0_|d 3_x0 Φ3= Z ρ0Π0 |~x − ~x0_|d 3_x0 Φ4= Z _p0 |~x − ~x0|d 3_x0 A = Z ρ0[~v0· (~x − ~x0)]2 |~x − ~x03_| d 3_x0 B = Z ρ0 |~x − ~x0_|(~x − ~x 0_{) ·}d~v0 dt d 3_x0

At this point we can change the gauge, in order to put metric tensor in a more convenient way. The gauge we choose is called standard Post-Newtonian gauge, is the one in which spatial part of metric tensor is diagonal and isotropic and in which g00 does not contain the term

B. We have found a very general form for the Post-Newtonian metric in any metric theory of gravity. This general theory is called parametrized Post-Newtonian (PPN) formalism and the parameters are called PPN parameters. The only way that the metric of a theory can differ from another theory is in the value of the coefficients that multiply each term of the metric. Thus we put an arbitrary PPN parameter in front of each Post-Newtonian term of the metric. We denote the parameters γ, β, ξ, α1, α2, α3, ζ1, ζ2, ζ3, ζ4. So the PPN metric is

g00 = −1 + 2U − 2βU2− 2ξΦW + (2γ + 2 + α3+ ζ1− 2ξ)Φ1 +2(3γ − 2β + 1 + ζ2+ ξ)Φ2+ 2(1 + ζ3)Φ3 +2(3γ + 3ζ4− 2ξ)Φ4− (ζ1− 2ξ)A g0j = − 1 2(4γ + 3 + α1− α2+ ζ1− 2ξ)Vj − 1 2(1 + α2− ζ1+ 2ξ)Wj gij = (1 + 2γU )δij (2.5)

The parameters ζi and αi are strictly related to conservation laws of energy, total momentum

and the existence of a preferred reference frame, for more details see [Will 1993]. These facts allow us to divide Post-Newtonian theories in three classes

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PPN parameters

{ζ₁, ζ2, ζ3, ζ4, α3} {α1, α2} Type of theory

all zero all zero Fully conservative all zero may be nonzero Semiconservative may be nonzero any value Nonconservative

It is possible to give a sort of physical significance to PPN parameters, but it can not be considered as a covariant statement, because it is valid only in the standard PPN gauge.

PPN parameter What it measures

γ How much space-curvature is produced by unit rest mass?

β How much nonlinearity is there in the superposition law for gravity? ξ Are there preferred-location effects?

α1 α2 α3   

Are there preferred-frame effects? α3 ζ1 ζ2 ζ3 ζ4           

Is there violation of conservation of total momentum?

Perhaps it will be more useful to have also a form of the metric tensor that take in account that the Post-Newtonian system could move respect to the rest universe, which is what occurs for the Solar System. It is possible to find this form of the metric tensor, making a Lorentz transformation from the reference frame used until now and then a suitable change of gauge, for details see [Will 1993]. If ~w is the velocity relative to the old frame, we have that the metric tensor becomes g00 = −1 + 2U − 2βU2− 2ξΦW + (2γ + 2 + α3+ ζ1− 2ξ)Φ1 +2(3γ − 2β + 1 + ζ2+ ξ)Φ2+ 2(1 + ζ3)Φ3 +2(3γ + 3ζ4− 2ξ)Φ4− (ζ1− 2ξ)A − (α1− α2− α3)w2U −α2wiwjUij+ (2α3− α1)w1Vi g0j = − 1 2(4γ + 3 + α1− α2+ ζ1− 2ξ)Vj− 1 2(1 + α2− ζ1+ 2ξ)Wj −α₂wjUij − 1 2(α1− 2α2)w i_U gij = (1 + 2γU )δij . (2.6)

What we found is a very general metric theory of gravity in the limit of weak field and slow motion depending on parameters which have to be determined. Given some values to the PPN parameters we obtain a possible metric theory. Conversely every gravitation metric theory has its Post-Newtonian formulation, i.e. has specific set of values for the PPN parameters. We will see in next section how to find Post-Newtonian form of any metric theory of gravity.

2.1.2 Post-Newtonian limit

In the previous section we have found a general metric theory for gravitation that depends on ten parameters, and giving them a set of values, it includes all possible gravitational metric theories. Now we introduce a general method for calculate the Post-Newtonian limit of any metric theory, i.e. we see how to calculate PPN parameters for a given gravitational metric theory. This process is divided into the following 9 steps.

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Step 1: Identify the variables, which may include dynamical variables such as a metric gµν, a

scalar field φ,a vector field Kµ and a tensor field Bµν, some prior-geometrical variables

such as cosmic time t and a flat background metric ηµν, and matter and nongravitational

variables.

Step 2: Set the cosmological boundary conditions. This means, assuming homogeneous coor-dinates, define the values of the variables far from the system at a chosen moment of time (asymptotic value). With these conditions a convenient choice, compatible with the symmetry of the situation, is, for the dynamical variables,

gµν → g(0)µν = diag(−c0, c1, c2, c3),

φ → φ₀,

Kµ → Kµ(0)= (K, 0, 0, 0),

Bµν → Bµν(0) = diag(ω0, ω1, ω1, ω1),

and for the prior-geometric variables,

ηµν = diag(−1, 1, 1, 1)

T = T

Step 3: Expand in post-Newtonian series about the asymptotic values the variables gµν = gµν(0)+ hµν

φ = φ0+ ϕ

Kµ = Kµ(0)+ kµ= (K + k0, k1, k2, k3)

Bµν = Bµν(0)+ bµν.

Step 4: Substituting these forms into the field equations, keeping only such terms necessary to obtain a final, consistent Post-Newtonian solution hµν. For the matter sources use the

perfect fluid stress-tensor Tµν.

Step 5: Solve for h00 to O(2). Assuming that h00→ 0 far from the system, we obtain

h00= 2αU

where U is the Newtonian gravitational potential and α may be a complicated function involving cosmological matching parameters and other constants that may appear in the field’s equation. So to Newtonian order metric has the expression

g00= −c0+ 2αU, g0j = 0, gij = δijc1

If we make the transformation

x¯0 = 1 c0

x0, x¯j = 1 c1

xj, (2.7)

metric and gravitational potential transform like g¯_0¯₀= (1/c0) g00, g¯_0¯_j = 1/(c0c1)1/2 g0j, g¯_i¯_j = (1/c1) gij ¯ U = c1U