A Mathematical Introduction to General Relativity

The following sections are included:

• Dedication
• Preface
• Notation and terminology
• Contents

As mentioned in the preface, spacetime is, on the set-theoretic level, a collection of points, called events. We want to do calculus on spacetime, because firstly, the field equation determining the geometry of spacetime is a partial differential equation on spacetime, and secondly, the geodesic equation describing motion of matter in spacetime is an ordinary differential equation on spacetime. It was discovered in the early 20^th century that the appropriate structure to be used to describe spacetime is that of a Lorentzian smooth manifold. In this chapter, we will first learn about the notion of a smooth manifold.

Roughly speaking, an m-dimensional smooth manifold is a set M which can be covered by patches such that in each patch one can introduce coordinates and use them to do calculus. As coordinates are ad-hoc, we need to make sure that only certain 'admissible' coordinates are allowed, so that the definition of smoothness is independent of the choice of coordinates. It turns out that this also endows the manifold with a topology, so that smooth manifolds are special types of topological spaces.

Intuitively, a tangent space at a point p on a surface M in ℝ³ is the plane at p tangential to the surface, consisting of all tangent vectors. Tangent vectors are the ‘velocities’ of curves passing through p. Thus we imagine a picture like this: Tangent plane T_pM and tangent vector v at a point p on the surface M. However, when we have only an abstract manifold M at hand, this is no longer an adequate definition, since there is no ‘natural’ embedding in ℝ^d. We ought to keep the definitions ‘intrinsic’, not relying on any embedding, and using just the manifold structure. The way around the obstacle is that we begin by revisiting the familiar notion above, and express it in a way which lends itself to an appropriate generalisation…

In the previous chapter, we defined the notion of a tangent vector. We now want to talk about ‘(smooth) vector fields’. Intuitively, a vector field on a manifold is thought of as a distribution of tangent vectors on the manifold so that they change ‘smoothly’ from point to point. The wind velocity on the surface of the Earth is an example of a vector field. But we wish to define vector fields ‘intrinsically’ (i.e., built using the abstract manifold), and moreover, the vector field should be ‘smooth’ (to do calculus)…

In this chapter, on smooth manifolds M, using the building blocks $T_0^{1}M$ and $T_1^{0}M$, namely the C^∞(M)-modules of all vector fields and of all 1-form fields, we will introduce (r, s)-tensor fields, which will be C^∞(M)-multilinear maps defined on the Cartesian products of these modules to C^∞(M). As we had mentioned earlier, tensor fields are fundamental in Lorentzian geometry and in spacetime physics. The Lorentzian geometry will be specified by the metric g on a smooth manifold, where the metric will be a certain type of a (0, 2)-tensor field. This metric will induce a curvature tensor field R (via something called the Levi-Civita connection), and R will be a (1, 3)-tensor field. Vector field and 1-form fields are themselves simple examples of tensor fields: vector fields are (1, 0)-tensor fields, and 1-form fields are (0, 1)-tensor fields.

So far, we have introduced notions using only the smooth structure on the manifold. But the smooth structure is still rather flexible. Within this picture, two spheres in ℝ³ of different radii would be considered the same, and even the surface of a potato would be the same smooth manifold as a sphere. In this sense, the smooth manifold has not yet acquired a ‘shape’. We now introduce the notion of a metric, which is roughly speaking, a (0, 2)-tensor field g on a manifold M, whose evaluation at any point p ϵ M equips the tangent space T_pM with a scalar product g(p) (analogous to the dot product of vectors in the plane). This structure allows us to assign lengths to tangent vectors and angles between them. It will also enable us to assign lengths to curves…

In this chapter, we will introduce additional structure on a smooth manifold, called a connection. This will allow us to calculate the directional derivative of a vector field in a given direction. While there are infinitely many connections definable on a smooth manifold, in the context of a semi-Riemannian manifold, there is a certain natural one, called the Levi-Civita connection induced by the metric g. For spacetimes (M, g), we will always use the Levi-Civita connection induced by g…

We will now learn that a connection induces a way of ‘parallel transporting’ vectors along a curve γ. So if p = γ(a) and q = γ(b) are points along the curve γ, then $P_{ab}^{\gamma }:T_{p}M\to T_{q}M$ will be a linear map which will transport vectors from T_pM to T_qM ‘parallelly’ along the curve. We will explain the meaning of this below. In this manner tangent spaces at different points can now be ‘connected’, and this is the motivation behind the name ‘connection’. We will also see that if M is a semi-Riemannian manifold, with ∇ taken as the Levi-Civita connection, then the parallel transport map is an isometry.

In the previous chapter, we learnt that a connection ∇ on a smooth manifold M allows us to differentiate a vector field W along a curve γ, using the induced operator ${\nabla }_{V_{\gamma }}$, where V_γ is the velocity vector field along the curve. Thus, we can now talk about the acceleration along a curve by taking W = V_γ itself, so that ${\nabla }_{V_{\gamma }}{V}_{\gamma }$ measures how fast V_γ changes as we move in the direction of V_γ, which exactly coincides with our intuitive notion of acceleration. If the acceleration everywhere along the curve is zero, then we call the curve a ‘geodesic’. In the spacetime context, geodesics are the worldlines of ‘freely falling observers’.

Geometrically speaking, geodesics are the 'straightest possible curves' in a smooth manifold M with a connection ∇, and are the generalisation of straight lines from the flat Euclidean plane case. However, we will also see that we should not think of geodesics $\gamma : I \to M$ merely in terms of the image _γ(I) of points it describes, but the parametrisation is crucial, and a reparametrisation of a geodesic may no longer be a geodesic. Roughly speaking, only affine linear maps of the arc-length parametrisation are admissible for maintaining the geodesic nature of a curve.

The extent to which the commutator of covariant derivatives differs from the covariant derivative along the commutator of the vector field is used as a measure of the ‘curvedness’ of a manifold with a connection. In this chapter, we will learn about this object, namely a (1, 3)-tensor field R, called the curvature tensor field defined as follows: for vector fields $X,Y,Z\in T_0^{1}M$ and a 1-form field $\text{Ω}\in T_1^{0}M$,

We had met the curvature tensor field R in Exercise 6.20 (p.122), where we had seen that for a locally flat semi-Riemannian manifold (M, g), the curvature tensor field R is zero. For example, R is zero for the Minkowski and cylindrical spacetimes, as these are locally flat. But R ≠ 0 for S² considered as a Riemannian manifold with the metric induced on its tangent spaces from the Euclidean inner product on ℝ³ (Exercise 6.20, p.122). In the appendix to this chapter we will also learn that if R = 0, then the semi-Riemannian manifold is locally flat. This provides the motivation for considering R to be a quantification of curvature. We begin with the following definition…

In this chapter, we will discuss special types of tensor fields on M, called ‘form fields’ (a special but trivial example of which is a 1-form field), and develop a useful calculus of these, after introducing the ‘exterior derivative’. This will enable us to write Maxwell’s equations of electromagnetism succinctly in a later chapter. The exterior derivative generalises the grad-div-curl vector calculus language from ℝ³ to smooth manifolds.

The language of form fields will also allow us to talk in particular about a 'volume form field' on a Lorentzian manifold, enabling a coordinate-free manner of integrating functions on a manifold. The simplest form field is a 1-form field. A one form field can be integrated along a curve: at any point of the curve, the point evaluation of the 1-form field on the tangent vector to the curve produces a number, and we add these contributions along the curve. The higher k-forms can be used to integrate along higher dimensional 'submanifolds', but in this book we will only consider the full manifold case and integrate m-forms on m-dimensional manifolds.

In the previous chapter, we learnt about k-form fields on smooth manifolds. We will learn in this chapter that these are the objects which can be integrated over ‘oriented’ manifolds. If we want an intrinsic notion of integration, we will need the notion of ‘orientation’. The root cause of this is that when performing the usual Riemann integral on ℝ, there is a subconscious choice made of orientation, namely

In this chapter, we will study Minkowski spacetime physics which served as a precursor to the development of the geometric theory of gravitation. At the end of the 19^th century, it was realised that the classical (Newtonian/Galilean) viewpoint of space and time was inadequate, as it could not explain experimental observations involving the measurement of the ‘speed of light’ c. The revision of the notions of space and time with the 1905 ‘special relativity theory’ by Einstein gave rise to the simplest spacetime, namely Minkowski spacetime, which we will discuss in this chapter. But before doing so, we quickly recall the familiar classical viewpoint, in order to be able to contrast it relatively easily with the Minkowski spacetime.

We remark that most of the definitions and results in this chapter pertain to objects in the tangent space T_pM of a general time-oriented Lorentzian manifold M at a point p. Even in the case of Lorentzian manifolds, the local region around a point resembles Minkowski spacetime, and the exponential map exp_p maps a neighbourhood $U \subset T_{p}M$ of 0 onto a neighbourhood exp_p U of p in M.

As we had mentioned in the preface, the field equation is $${}_{\left(\text{geometry}\right)}\mathbf{\text{Ric}}-\frac{1}{2}\mathbf{\text{Sg}}\text{+}\mathbf{\Lambda }\mathbf{\text{g}}=8\pi \mathbf{\text{T}}{\text{. }}_{\left(\text{matter}\right)}$$ We now focus on the right hand-side of this equation, and also provide some motivation for this equation. (The real belief in the equation as the governing equation for spacetime rests on its ability to predict physical phenomena, and we have already met two of these tests, namely the red-shift, and the deflection of light. We will also meet a third test, on the perihelion shift of the orbit of Mercury, where the observed discrepancy with the Newtonian prediction was correctly accounted for within the spacetime geometric view of gravitation.) We begin by revisiting Newtonian gravity, and recalling the Poisson equation that shows how the matter sources determine the ‘gravitational potential’. This serves as a precursor to the field equation above.

In the last chapter, we saw that the energy-momentum tensor field T on a spacetime (M, g) is assumed to satisfy

In Section 14.3, we saw that the smooth manifold M = ℝ × (r_min, ∞) × S², where r_min = 2m > 0, is a spacetime with the Schwarzschild metric g. In a spherical coordinate chart described there, g is given by

In this chapter, we will see that in the spacetime-geometry based theory of gravitation, the field equation is able to give information about the universe as a whole. To achieve this, we ignore the local details (planets, stars, galaxies, etc.), and instead adopt a global view by making reasonable assumptions about the energy-momentum tensor field T in the ‘large scale’. This is akin to how we think of a fluid when we model its flow, ignoring the individual molecules or particles in the fluid. In smoothed-and-averaged form, we will assume that the energy-momentum tensor field is that of a perfect fluid. The picture below is meant to indicate the scales of distances in the visible universe…

The following sections is included:

• Notation and Terminology

The following sections are included:

• Notes
• Bibliography
• Index