The geometry of relativity

General Relativity has always fascinated me; it is probably one of the main reasons why I decided to study mathematics and physics, and it is definitely an important reason for studying Differential Geometry and Geometric Analysis, even if my research is not focused in Einstein's equations. Why is it so fascinating? Well, that's what I am going to explain on Friday, in the Graduate Student Seminar (1pm-2pm, Surge 284, UCR). But in case you can not make it, let me explain you a few things about relativity.

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By the end of the XIXth century, if you were a physics major you would be studying:

1. Newtonian Mechanics (Galilean transformations, inertial observers...)
2. Electromagnetism (Maxwell's equations)
3. Thermodynamics / Classical statistical mechanics

However, theories 1 & 2 are not compatible, at least from a philosophical (thus fundational) point of view. The foundational principles of Newtonian Mechanics assert that there exists a collection of special observers, called inertial observers, with respect to which the laws of physics are invariant. Any of these observers will be able to describe a physical phenomena using the same laws, and there is a group of transformations between observers that preserve these laws (these are called the Galilean transformations, and include translations, rotations, symmetries, and observers moving away at a constant speed). The Galilean transformations leave invariant the 2nd law of Newton F=ma (in fact, by definition, they are any transformation that leaves invariant this equation), but it turns out that they do not leave invariant Maxwell's equations. In fact, Maxwell's equations predict that the speed of light is constant, so there is no way to allow this to happen and, at the same time, allow two observers moving at some constant relative speed to be inertial observers.

Hence, in the beginning of the XXth century, physicists were trying to reformulate either one of the two theories, in order to make them compatible. Einstein decided to modify Newtonian Mechanics, as shown in his first famous paper of 1905; to do so, he had to forget about the idea absolute time and space (common for all the inertial observers), which are very natural ideas for the human brain (it seems reasonable that two inertial observers should measure the same amount of time between two events, and the same length between two points). But how can one do physics if two observers do not measure the same length intervals or times between two events? Well, using mental experiments, Einstein build his theory of Special Relativity from the following postulates:

1. The laws of physics are the same for any inertial observer.
2. The velocity of light, c, is a constant of nature.
3. Space & time are not independent, they are together in a 4D space called spacetime.

From the 2nd postulate, it follows that the time it takes for a beam of light to travel between two events is the same for any two inertial observers; hence, they will always agree on the quantity ds² := -c²dt²+dx²+dy²+dz². This suggests to model spacetime with a 4D vector space together with the bilinear form ds², which we will consider as some sort of "dot product" (which provides us with some notion of "distance" and "length"); this space is known as the Minkowski spacetime. The transformations relating inertial observers, called the Lorentz group, is the set of linear transformations that leave invariant the form ds².

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Notice that the bilinear form, usually denoted by η, is symmetric but it is not positive definite! Hence, it is not defining a dot product, nor a metric in the space; however, it is non-degenerate, so we say that it defines a pseudo-metric. In this space, there are vectors with zero norm which are not the zero vector (those relating two events that can only be connected by an observer travelling at the speed of light), or even vectors with negative length (those between two events that can not be reached by any observer travelling at a velocity below or equal to the speed of light).

However, Special Relativity has several weaknesses:

• It does not explain gravity: Newtonian gravity involves a vector field connecting instantaneously spacelike events (events separated by negative distances).
• How do we distinguish, a priori, between an inertial frame and a non-inertial frame?

Inspired by the equality between the inertial mass (the one in the formula F=ma) and the gravitational mass (the one that gives us the gravitational force mg), Einstein realized that there can not be a theory of gravity without a theory that allows observers that are being accelerated. Hence, he decided to build the Theory of General Relativity starting from the Equivalence Principle, that states that the laws of physics must hold for any observer (not only for a specific class of observers, namely the inertial ones); in other words, gravity is equivalent to accelerated frames, it is a geometric property of spacetime, and any motion described by a freely falling object (that might be accelerating with respect to us) is due to a gravitational field. This removes the philosophical problem that all the previous theories had, about distinguishing a priori between inertial and non-inertial observers. It also allows to build a much more geometric physical theory, were "arbitrary" (or smooth enough) transformations between observers are allowed; the requirement of the transformations between observers being isometries (transformations preserving η) vanishes.

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Allowing arbitrary transformations implies that the velocity of light is not constant anymore, nor are the coefficients of the bilinear form η. However, locally we expect the General Theory of Relativity to reproduce the Special Theory of Relativity, so there is some sort of bilinear form at each point, that changes from point to point, but that in the right coordinates looks, locally, like η. That is to say, our spacetime is a manifold (we no longer require a flat spacetime), together with a pseudo-metric: an inner product (pseudo-dot product) that changes smoothly from point to point. This object is called a pseudo-Riemannian manifold, and in General Relativity spacetime is modelled by a 4D pseudo-Riemannian manifold.

So we have a pseudo-Riemannian manifold, whose geometry determines all the gravitational effects. We know that in Newtonian gravity there is a relationship between the mass distribution and the gravitational field in the space, so since we expect to reproduce Newtonian gravity at small scales, that means that there should be an equation relating the mass (or energy, equivalently) distribution in spacetime to the geometry. What kind of equations should we look for? Since they have to agree with Newton's theory of gravity, they should be second order differential equations involving, for example, vector fields. Moreover, according to the Equivalence Principle, they must not depend on coordinates (as physicists say, they must be covariant); this is very easy to understand from the point of view of Differential Geometry: the equation should be an equality between sections of some vector bundle, thus independent of the chosen coordinates (for example, an equality between vector fields, as defined on a manifold). More generally than just vector fields, we look for an equation between tensor fields.

On the left hand side we will have a tensor coming from geometry (the Einstein tensor), and on the right hand side we will have a tensor with the physical information (the Energy-Momentum tensor, already known in fluid mechanics). Einstein's tensor is constructed from Ricci's curvature tensor, the scalar curvature, and the metric. An additional term, proportional to the metric, can be added, but it is not considered as part as Einstein's tensor, since he refused to add this term, that would have as a consequence a non-static universe (if he had considered it, he would have been able to predict the expansion of the universe). Summarizing, Einstein's field equation looks as follows:

Here the unknowns are the metric coefficients, while the Energy-Momentum tensor T is some given data (in the same way that the force F in F=ma is given by the problem that we are studying); all the other tensors are constructed in terms of the pseudo-metric g. This system of equations is a very non-linear system of 10 PDEs; even when studying empty spacetime (T=0), and considering only the first two terms in the left hand side, solving the system seems really difficult! However, surprisingly, only a month after the publication of the equations by Einstein, Schwarzschild published one of the most useful solutions to these equations: the Schwarzschild solution for rotationally symmetric spaces in vacuum. This solution can be applied to understand the motion of the Earth around the Sun, by neglecting the mass of the Earth and considering that all the curvature comes from the Sun; the Earth is a freely falling massless body, hence it follows one of the geodesics. Unfortunately, the two-body problem is still an open problem in General Relativity, and it can only be solved numerically. But so far, this solution was enough to locate General Relativity in the center of modern large scale physics.