General relativity is based on three fundamental principles, the general principle , the special principle, and the principle of equivalence, together with the empirical fact that we establish coordinate systems locally through the physical measurement of time and position. The proper application of these principles requires a detailed understanding of mathematics. This page will discuss these ideas and their implication, and will motivate the mathematical treatments which follow.
the principle of uniformity in nature,
The principle of uniformity in nature: The fundamental behaviour of matter is always and everywhere the same
together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy) and the empirical fact that we do not have a universe obeying Newtonian relativity. These notions are encapsulated in the general principle of relativity,
The general principle of relativity: Local laws of physics are the same irrespective of the reference matter which a particular observer uses to quantify them.
The principle of general covariance is the mathematical implementation of the general principle of relativity. In non-mathematical language it says “local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”. Vectors are not invariant, as their coordinate representation changes with the coordinate system, but relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be covariant. Tensors are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,
The principle of general covariance: The equations describing local laws of physics have tensorial form.
Despite the extreme reasonableness of this assumption, it is extraordinary in its strength. It appears that the mathematical constraints it places on physical laws are so strong that the only possible theory of gravity or of the structure of space and time is the geometrical theory described by general relativity. As will be seen in the mathematical development, general covariance determines the whole of the theory of general relativity, while the special principle and the equivalence principle relate mathematical structure to physical observation.
Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K. —Albert Einstein: The foundation of the general theory of relativity
The principle of relativity was first explicitly stated by Galileo, using an argument known as Galileo’s ship, which he also tested by dropping objects from the mast of a moving ship. In special relativity, Einstein extended the application of the principle in two ways, using it to establish the coordinate system and the constancy of the speed of light.
With general relavity, Einstein further extended the special principle, formulating the general principle and the principle of general covariance, but the special principle remains significant, as the means by which we use Newton’s laws to identify inertial frames as defining a special class of coordinate systems in which there is a natural correspondence between mathematical structure and physical behaviour.
N1*: An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matterIn a geometrical theory of spacetime, Newton’s first law determines uniform motion of inertial matter locally within an inertial reference frame. Since objects always move in the direction of their velocity vectore, the immediate consequence is geodesic motion, meaning that gravity is not an active force, in the sense of Newton. Although the equivalence principle was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a g-force experienced by an observer in an accelerated frame of reference.
The Equivalence Principle: We ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system. (Einstein 1907).
Minkowski Coordinates, but there is no guarantee that a mapping of distant points to these coordinates can be made without distortion of the map. The situation is analogous to mapping the surface of the Earth. At any point of the Earth’s surface, a cartographer can make a locally flat map. He cannot extend the map without distortion, but this does not mean that geometry at other points of the Earth surface is different from the geometry seen by the cartographer.
Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called manifolds. Spacetime is described as a Lorentzian manifold. By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the Pound-Rebka experiment. In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys special relativity. Together with Einstein’s field equation, this determines a curved geometry which precisely accounts for Newton’s law of gravity.
Definition: A manifold is a structure in which any point has a neighbourhood which can be described by a coordinate system or chart.
Typically a single coordinate system cannot be used to give a full description of a manifold. A collection of charts which describes the whole manifold is an atlas.
atlas, or collection of charts. In common with many definitions of mathematical structures, this does not tell us what the manifold actually is. Instead it tells us what properties a manifold has, how a manifold behaves.
If we have some object, and we can show that that object has the properties of a manifold, then we can say that the object is a manifold. We do this, for example, when we say that the surface of a sphere is a manifold. However, to do that we must first have the object. We know how the manifold behaves in general relativity, but it is natural to pose the question, "What is spacetime?". I know of three possible answers:
Answer 1. There exists a substantive spacetime, which is modelled (i.e. described) by the mathematical structure of a manifold.
Here the manifold simply replaces Newton’s conception of absolute space and absolute time. This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We observe the behaviour of matter, and infer the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.
The logical error lies in thinking that if we have a set of actual observations, B, and a theory, A with A => B, then A must be true. In fact, there may be some other theory, C, which we may not know about, which also has C => B, and such that C contradicts A. Modern physicists usually avoid the issue by denying that it is possible to describe nature:
Answer 2. The mathematical structures of physics do not model anything. They are just algorithms whose validity rests only on correspondence between prediction and experiment.
This position can be regarded as the current orthodoxy. It is adopted by many theoretical physicists, especially quantum field theorists. There is no arguing with it. It is both as solid, and as absurd, as schoolboy solipsism. It is clear to me that realism is a prerequisite for science as a meaningful activity, and that when I say "there is a tree in the park" I am describing reality. Our ability to describe a tree in the park refutes, at a very obvious level, the proposition that reality cannot be described. I always find it surprising when physicists advocate the idea that reality cannot be described, because it undermines the very purpose of physics. In fact, Einstein has already refuted the idea that science requires us to infer theory from agreement between prediction and experiment. Special relativity is based on based on empirically verifiable postulates, the operational definitions of measurement. Einstein’s argument is the archetype according to which we should formulate modern scientific theory, starting with how we define the numerical quantities which we use in the scientific study of nature. Special relativity is imported, as local theory, into general relativity, and provides the basis for understanding what spacetime actually is:
Answer 3. I observe that I can, in principle, choose reference matter anywhere I wish, and that I can define Minkowski coordinates relative to that matter. I now define spacetime by imagining all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.
“A physical quantity is defined by the series of operations and calculations of which it is the result” — Sir Arthur Stanley Eddington, The Mathematical Theory of Relativity, 2nd ed., p. 3, 1923Only measurements which are actually carried out have physical reality, and generate coordinates for physical events, whereas spacetime consists of all the ways in which this can conceivably be done. So, spacetime does not model reality — only the small subset of spacetime for which there are actual measurements corresponds to reality. Spacetime models our conception of reality, not reality itself. At the same time, spacetime does contain the real observational results required for comparison between theory and experiments.
Position 3 avoids both the unjustified metaphysical assumption of answer 1 and the feeble abdication of the purpose of science inherent in answer 2. Instead it correctly bases science on observation. The proper use of mathematics will enable predictions to be rigorously deduced, not inferred by induction. Modern science can thus escape the problem that a premise cannot be deduced from a conclusion.
general covariance automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in special relativity, using the radar method. In this case the chart is made on Minkowski spacetime, which has constant Minkowski metric, h. h is a non-physical metric, analogous to the metric of the paper on which a map is drawn. h does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a tangent chart, an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.
Definition: A tangent chart is defined to use coordinates as defined by an inertial observer using the radar method.
For an inertial observer, metre sticks give the same result, but I have elected to use radar because it leads to a simpler analysis, and is applicable to measurements generally in the Solar System. A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be transformed into coordinates with Minkowski metric at the position of the observer.
Definition: A Proper quantity is a physical quantities as it would be measured in Minkowski coordinates by an inertial observer moving with that quantity.
Definition: Proper time is the amount of time for an object, as it would be measured on a local clock moving with that object.
Definition: Proper length is the length an object as it would be measured by an observer moving with that object.
According to general covariance physical quantities are scalars, vectors and tensors. Vectors and tensors have different representations, dependent on the coordinate system being used. A proper quantity is the representation of a physical quantity as seen in locally Minkowski coordinates moving with the quantity being measured.
C, of the orbit, that is the sum of the lengths of small sections as they would be measured by stationary observers at each point. Similarly the proper radius, R, is sum of the lengths of small sections along a radius as they would be measured by stationary observers at each point on the radius.
It is observed, for example using clocks on GPS satellites, that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the equivalence principle and the Pound-Rebka experiment. Consequently, relative to coordinates determined by radar, proper distances at the surface of the planet are greater than proper distances in orbit (to preserve the speed of light which is used for the definition of distances locally). Consequently the proper length of the circumference is greater than 2π times the proper radius, C > 2πR. This means that space has a curved geometry in the region of a gravitating body.
|Proper radius, R, is not easily measured, and is not a convenient quantity for the definition of coordinates. A commonly used choice is to define the Schwarzschild radial coordinate, r, such that r = C ⁄ 2π.|
Definition: The Schwarzschild radial coordinate r is given by the proper arc length l of a small arc on a sphere with centre at the origin divided by the angle dθ subtended by that arc: r = l ⁄ dθ = C ⁄ 2π
Definition: Schwarzschild coordinates have spacetime metric given by:
where k and κ are functions of position, x. k(x) is the redshift of light passing from the origin to x.
It is is necessary to solve Einstein’s equation to determine the functions k and κ.
orthogonal. In this case the metric is diagonal (as seen in Schwarzschild coordinates) and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.
A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the lensed and mirrored geometries, which are actually flat. To describe curvature requires a connection in addition to the metric field. Given the metric field, an affine connection describes a relationship between a set of coordinate axes at x, say, and another set, at x +dx, where dx is a small displacement, such that we can meaningfully describe a vector at x as being parallel to one at x +dx (other types of connection are used to transport other types of data).
In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between vectors at nearby points using parallel displacement in tangent space, and projecting back into the curved surface. This is the Levi-Civita connection, defined in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through parallel transport, and is path dependent. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.
chart of spacetime need not be a physical map, like the maps in a world atlas. A mathematical idealisation suffices just as well — that is to say, the map may consist of tables of data and/or formulae. We may imagine, for example, the numbers, or coordinates, describing the times and positions of physical events mapped into a bank of computer memory. In principle, using a large enough bank of computer memory, this could be done to any precision, for as many points as one requires, and a map of a region of spacetime could be produced with any required level of detail, up to the limit of accuracy of measurement and the size of available computer memory.
In general relativity, spacetime is described as a differentiable manifold. This means that we assume that we may apply the mathematical operation of differentiation to functions defined on the mapping space and we claim that differentiation is physically meaningful when those functions correspond to measurable physical quantities. For example, in classical physics we regularly differentiate a velocity to find an acceleration, or a potential to find a force. In the strictest sense, a map which consists of tables of data, or a bank of computer memory, is not continuous and is not differentiable. In computer science derivatives are regularly approximated with difference relations. Similarly, in applications to physics, differentiability does not require taking a mathematical limit, but merely requires a step size small enough that any errors are smaller than measurement errors and taking it smaller makes no practical difference to calculation. General relativity is a theory of classical matter, and is used to study large scale phenomena, anything from the gravity of a planet to the large scale structure of the universe. On such scales the assumption of differentiability is entirely reasonable and leads to no practical issues or disagreement between theory and observation. It will be necessary to revisit questions of continuity and differentiability in the study of quantum phenomena.
The General Theory of Relativity ↑ Mathematical Methods →