Lecture Five: Theory of Relativity

5.1 Construction of a theory of gravity

We need some guiding principles to construct a mathematical theory of gravity. As I have explained in the lectures, a number of lines of reasoning led Einstein to make the following assumptions. These assumptions are sufficient to define a theory of gravity (though as we will see they do not define a unique theory).

• The Principle of Equivalence

In any gravitational field we can choose a local freely falling inertial frame in which the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in the absence of gravity.

• The laws of Physics are covariant

This means the following:

∘ We can find a transformation from a general coordinate system x^{μ} to a locally inertial system ξ^{μ}

\frac{∂ξ^{α}}{∂x^{μ}} .

(These transformations will be non-linear in general).

∘ The equations of physics must agree with the laws of Special Relativity in the absence of gravitation.

∘ Physical laws must have a tensorial form, so that they can be stated in a form that is invariant under a general coordinate transformation x^{μ} → x^{′μ}.

• Riemannian geometry

We need a mathematical description of non-Cartesian geometry for spaces described by a metric

ds^{2} = g_{μν}dx^{μ}dx^{ν}.

How do we proceed? We could start with Newtonian gravity, and try to make it consistent with the relativity principle.

The familiar equation of motion in Newtonian gravity is

\frac{d^{2}x}{dt^{2}} = −∇Φ,

(which is not Lorentz invariant), where the gravitational potential Φ is determined by Poisson’s equation (the field equation of Newtonian gravity),

∇^{2}Φ = 4πGρ.

Notice that the ∇^{2} operator in Poisson’s equation is equivalent to the d’Alembertian operator in the limit that the speed of light tends to infinity

∇^{2} ≡  □^{2}
lim c → ∞

hence in Newton’s theory, the potential Φ responds instantaneously to a disturbance in ρ. Now, we could try replacing ∇^{2} by □^{2} in Poisson’s equation, and writing the equation of motion of particle in a relativistically invariant form. But, if we tried this, we would not end up with a consistent theory. In fact no consistent theory can be constructed in which the analogue of the gravitational field is a scalar or a vector. We will see that we are led to a (rank 2) tensor theory of gravity. This is quite a large leap from Newtonian theory and is motivated by non-Euclidean geometry. For this reason, many physicists talk about gravity as an ‘inherently’ geometrical force. I don’t agree with this and take the view that gravity is just another field theory that has a particular geometrical interpretation. In any case, whatever ones philosophical leanings, there is no question that gravity and geometry are closely intertwined and so it is to non-Euclidean geometry that we now turn.

5.2 Non-Euclidean geometry

All of you, even the physicists amongst you, have a working knowledge of non-Euclidean geometry. Even the ancient Greeks knew about non-Euclidean geometry. I will show you what I mean with a simple example.

Figure 1:

Let us imagine a usual Cartesian coordinate system x, y, z defining a Euclidean threedimensional space with line element

ds^{2} = dx^{2} + dy^{2} + dz^{2}.  (1)

Now, suppose that we have a sphere of radius a with its centre at the origin of our coordinate system as shown in Figure 1. We will now ask the following question: what is the line element (i.e. the analogue of equation 1) on the surface of the sphere?

The equation defining the sphere is

x^{2} + y^{2} + z^{2} = a^{2}.

So, differentiating this equation

2xdx + 2ydy + 2zdz = 0,  (2)

and so we can write an equation for dz,

dz = −\frac{(xdx + ydy)}{z} = \frac{−(xdx + ydy)}{[a^{2} − (x^{2} + y^{2})]^{1/2}} .

Equation (2) thus provides a constraint on dz that keeps us on the surface of the sphere if we are displaced by small amounts dx and dy from an arbitrary point on the sphere. Substituting for dz in equation (1) gives us the interval for such constrained displacements (in other words, the metric for the surface of the sphere)

ds^{2} = dx^{2} + dy^{2} + \frac{(xdx + ydy)^{2}}{[a^{2} − (x^{2} + y^{2})]}.

Now make the substitutions

\begin{aligned}x & = & r cos θ, \\ y & = & r sin θ, \end{aligned}

and after a little algebra we get

ds^{2} = \frac{a^{2}dr^{2}}{(a^{2} − r^{2})} + r^{2}dθ^{2},  (3)

(which, in the parlance of modern field theory, contains a ‘hidden symmetry’, namely our freedom to choose an arbitrary point on the sphere as the origin r = 0).

Note that this looks different to the metric we would write down using spherical polars

ds^{2} = a^{2}dθ^{2} + a^{2} sin^{2} θdϕ^{2},

but (3) is a valid metric nonetheless.

It is worth your while spending a little time to understand the metric (3) and its construction. Think about how the coordinates r and θ relate to points on the spherical surface. If you don’t understand this simple example, you don’t have a hope of understanding the more complicated example that follows. As a simple test of your understanding, let’s calculate the surface area of the sphere from the metric (3),

A = 2 ∫_{0}^{2π} ∫_{0}^{a} \frac{a}{(a^{2} − r^{2})^{1/2}} rdrdθ
= 4πa^{2},

i.e. the usual result for the surface area of a sphere. (Make sure that you understand where the factor of two comes from in front of the integrals).

We can make an analogous construction to find the metric for a three-sphere embedded in four dimensional Euclidean space (i.e. we want to find the metric on the surface of a four dimensional ‘sphere’).

The metric of the four dimensional Euclidean space is

ds^{2} = dx^{2} + dy^{2} + dz^{2} + dw^{2}  (4)

and, in analogy with the example above, the equation defining a three-sphere is

x^{2} + y^{2} + z^{2} + w^{2} = a^{2}.

Differentiating as before gives

2xdx + 2ydy + 2zdz + 2wdw = 0,

and so substituting for dw in (4) gives the equation for the metric

ds^{2} = dx^{2} + dy^{2} + dz^{2} + \frac{(xdx + ydy + zdz)^{2}}{[a^{2} − (x^{2} + y^{2} + z^{2})]}.

Transforming to spherical polar coordinates

x = r sin θ sin ϕ,
y = r sin θ cos ϕ,
z = r cos θ,

we get the metric

ds^{2} = \frac{a^{2}}{(a^{2} − r^{2})}dr^{2} + r^{2}dθ^{2} + r^{2} sin^{2} θdϕ^{2}.  (5)

Notice that in the limit a → ∞, the metric tends to the form

ds^{2} = dr^{2} + r^{2}dθ^{2} + r^{2} sin^{2} θdϕ^{2}

which is simply the metric of ordinary Euclidean three dimensional space

ds^{2} = dx^{2} + dy^{2} + dz^{2},

rewritten in spherical polar coordinates. The metric (5) therefore describes a non-Euclidean three dimensional space. Furthermore, the space is finite with a volume

V = 2 ∫_{0}^{a} ∫_{0}^{π/2} ∫_{0}^{2π} \frac{ar^{2}dr}{(a^{2} − r^{2})^{1/2}} sin θdθdϕ
= 2π^{2}a^{3}.

We can generate a metric for an infinite non-Euclidean three dimensional space by making the substitution a = ib, i.e. choosing the ‘radius’ of the space to be pure imaginary. The metric (5) then becomes

ds^{2} = \frac{dr^{2}}{(1 + r^{2}/b^{2})} + r^{2}dθ^{2} + r^{2} sin^{2} θdϕ^{2}.

5.3 Gauss’s hypothesis

In a sufficiently small region, we can construct a locally Euclidean coordinate system (ξ_{1}, ξ_{2}) such that the distance between two neighbouring points is

ds^{2} = dξ_{1}^{2} + dξ_{2}^{2}.  (6)

If a surface is not Euclidean, it will not be possible to cover any finite part of it with a Euclidean system ξ_{1}, ξ_{2}.

For a general coordinate system, (x_{1}, x_{2}),

dξ_{1} = (\frac{∂ξ_{1}}{∂x_{1}} ) dx_{1} + (\frac{∂ξ_{1}}{∂x_{2}} ) dx_{2},
dξ_{2} = (\frac{∂ξ_{2}}{∂x_{1}} ) dx_{1} + (\frac{∂ξ_{2}}{∂x_{2}} ) dx_{2}.

So, substituting into (6),

ds^{2} = g_{11} (x_{1}, x_{2}) dx_{1}^{2} + 2g_{12} (x_{1}, x_{2}) dx_{1}dx_{2} + g_{22} (x_{1}, x_{2}) dx_{2}^{2}

where

g_{11} = (\frac{∂ξ_{1}}{∂x_{1}} )^{2} + (\frac{∂ξ_{2}}{∂x_{1}} )^{2} ,
g_{12} = (\frac{∂ξ_{1}}{∂x_{1}} ) (\frac{∂ξ_{1}}{∂x_{2}} ) + (\frac{∂ξ_{2}}{∂x_{1}} ) (\frac{∂ξ_{2}}{∂x_{2}} ) ,
g_{22} = (\frac{∂ξ_{1}}{∂x_{2}} )^{2} + (\frac{∂ξ_{2}}{∂x_{2}} )^{2} .

Transforming from coordinates (x_{1}, x_{2}) → (x′_{1}, x′_{2}) will change the form of the metric functions g_{ij}, e.g., for the component g_{11}

g′_{11} = (\frac{∂ξ_{1}}{∂x′_{1}} )^{2} + (\frac{∂ξ_{2}}{∂x′_{2}} )^{2}
= (\frac{∂ξ_{1}}{∂x_{1}} \frac{∂x_{1}}{∂x′_{1}} + \frac{∂ξ_{1}}{∂x_{2}} \frac{∂x_{2}}{∂x′_{2}} )^{2} + (\frac{∂ξ_{2}}{∂x_{1}} \frac{∂x_{1}}{∂x′_{1}} + \frac{∂ξ_{2}}{∂x_{2}} \frac{∂x_{2}}{∂x′_{1}} )^{2}
= g_{11} (\frac{∂x_{1}}{∂x′_{1}} )^{2} + 2g_{12} \frac{∂x_{1}}{∂x′_{1}} \frac{∂x_{2}}{∂x′_{1}} + g_{22} (\frac{∂x_{2}}{∂x′_{1}} )^{2} .

Can we find a quantity that depends on g_{ij} and its derivatives and provides a measure of derivation from Euclidean geometry, independent of the coordinate system?

This problem was solved by Gauss in 1827 for two dimensional surfaces. Gauss discovered what is now known as the Gaussian curvature:

K(x_{1}, x_{2}) = \frac{1}{2g} [2\frac{∂^{2}g_{12}}{∂x_{1}∂x_{2}} − \frac{∂^{2}g_{11}}{∂x_{2}^{2}} − \frac{∂^{2}g_{22}}{∂x_{1}^{2}} ] − \frac{g_{22}}{4g^{2}} [(\frac{∂g_{11}}{∂x_{1}} ) (2\frac{∂g_{12}}{∂x_{2}} − \frac{∂g_{22}}{∂x_{1}} ) − (\frac{∂g_{11}}{∂x_{2}} )^{2}]

+ \frac{g_{12}}{4g^{2}} [(\frac{∂g_{11}}{∂x_{1}} ) (\frac{∂g_{22}}{∂x_{2}} ) − 2 (\frac{∂g_{11}}{∂x_{2}} ) (\frac{∂g_{22}}{∂x_{2}} ) + (2\frac{∂g_{12}}{∂x_{1}} − \frac{∂g_{11}}{∂x_{2}} ) (2\frac{∂g_{12}}{∂x_{2}} − \frac{∂g_{22}}{∂x_{1}} )]

− \frac{g_{11}}{4g^{2}} [(\frac{∂g_{22}}{∂x_{2}} ) (2\frac{∂g_{12}}{∂x_{1}} − \frac{∂g_{11}}{∂x_{2}} ) − (\frac{∂g_{22}}{∂x_{1}} )^{2}] ,

where

g(x_{1}, x_{2}) = det|g_{ij}| = g_{11}g_{22} − g_{12}^{2}.

If we evaluate this expression for the surface of a sphere, we find that the Gaussian curvature is K = 1/a^{2}. It is no real surprise that the curvature of a highly symmetric space such as a spherical surface is related to the radius a. But Gauss’s formulae gives a general relation for expressing the curvature of any arbitrary surface at any point. Where does Gauss’s formidably complicated formula come from? We will find out in later lectures – not only that, but we will generalize Gauss’s analysis to any number of dimensions.

In fact, in a space of D dimensions, there are

D(D + 1)/2

independent metric functions (g_{μν} is symmetric). The freedom to choose the functional forms of the D coordinates (which of course does not change the nature of the space itself) allows us to impose D functional relations on g_{μν}. This means that the space is characterised by

C = \frac{D(D + 1)}{2} − D = \frac{D(D − 1)}{2}

functions, so

D = 2 → C = 1
D = 3 → C = 3
D = 4 → C = 6

For D = 2, there is one function that expresses the ‘character’ or ‘inner’ property of the space – and this is the Gaussian curvature K. For three dimensions, we need to find three functions and so the problem is much more complicated. This formidable mathematical problem was solved by Riemann and we will need to understand some of the mathematics underlying Riemannian geometry before we can construct a consistent theory of gravity.

Before we do that, however, we can make a remarkable amount of progress by developing a formalism to describe freely falling particles and studying the motion of a particle in a weak gravitational field.