← The General Theory of Relativity ↑ →
In general relativity, Einstein put his physical ideas on the nature of time and space, into the mathematical language of tensors
and Riemannian geometry
The Spacetime Manifold
According to the general principle, an observer anywhere can use the radar method to define locally 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
What is Spacetime?
We can describe spacetime as a manifold, a geometrical structure which can be mapped onto an 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:
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
. 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:
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.
Only 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.
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.
In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, 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
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 by the radar method and has a constant, non-physical, Minkowski metric, equal to to the physical metric at an observer’s origin of coordinates.
Coordinate Time and Proper Time
In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer at a distance, and the same quantities as they would be described by a observer who determines them locally. Let Alf be an observer, with a clock at some point, A
, the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at B
, the origin of Beth’s coordinates. Alf and Beth both determine locally Minkowski Coordinates
. Alf’s time axis is denoted the 0-axis, and his space axes are labelled 1
. Beth’s coordinates are denoted with primes, 0'
Using general covariance
, each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing t'
seconds for a stationary object at the origin of Beth’s coordinates τ = τi' = (t', 0, 0, 0)
is the actual amount of time measured by Beth using her own clock, and is known as proper time
. In Alf’s coordinates,
is found by coordinate transformation
. If Beth is stationary in Alf’s coordinates, τ = τi' = (kt', 0, 0, 0)
, where k
is the gravitational redshift
factor, k = 1 + z
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.
The Spacetime Metric
, lowers the indices of contravariant
vectors in such a way that the inner product between vectors x
is an invariant
In particular, the magnitude |x|
of the vector x
so that the metric is a means of determining magnitude of a vector in any coordinate system.
The laws of physics local to Beth, as described by Beth, use proper times and distances determined in her local measurements. If Alf is to analyse physical processes close to Beth, he must also determine proper times and distances, using remote measurements. The definition of the metric ensures that, when Alf applies it to his own measurements, the magnitudes returned will be the proper times and distances of quantities local to Beth. General relativity allows that, in principle, Beth could be anywhere in Alf’s coordinate space. For each point, x
, where Beth could be, there is a different metric, gij(x)
Definition: The spacetime metric is defined, on a given coordinate system, by
The spacetime metric is often simply called the metric. One should avoid this abuse of language, because the spacetime metric is a tensor field
, that is to say it is a function having a different metric value at each point in spacetime. Calling it simply “the metric” confuses a metric field
, which is a function of coordinate space, with the metric
at a given position.
|In a static geometry, an observer, Alf, defines spherical coordinates by the radar method, using time t, determined from a clock at an origin at A. Coordinate distance, r*, from A is defined by setting the radial speed of light to unity. Spherical coordinates are orthogonal so that the spacetime metric is diagonal. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the spacetime metric has g00 = k2.|
|Beth determines proper distances local to B using the radar method, with lightspeed equal to unity. Since Beth’s clock runs faster than Alf’s, proper distances local to Beth are greater than corresponding coordinate distances in Alf’s coordinates by a factor k. Using unit light speed Beth calculates coordinate distance r = kr* to Alf.|
as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of B
, so that the angle subtended at A
by a small rod of proper length l
, perpendicular to AB
is l / r
, as it would be in flat space. This is the defining condition for Schwarzschild coordinates
Definition: The Schwarzschild radial coordinate is given by the proper arc length of a small arc on a sphere at the origin divided by the angle subtended by that arc. :
|In Schwarzschild coordinates a ring of short rods at radial distance r from A can be drawn on Beth’s map to form a continuous circle, centre A, without overlaps. It follows that, in spherical coordinates with origin A and radial distance r, g22 = −r2 and g33 = −r2sin2θ, and that, since Beth has increased the scale of local distances by a factor k, g11 = −k−2.|
Theorem: Schwarzschild coordinates in vacuum have spacetime metric given by:
where k and k are functions of positon, x, and k(x) is the redshift of light from the origin to x.
This result is usually found by solving Einstein’s field equation
. Here I have established it purely geometrically, because I think this gives greater insight and because it will simplify (slightly) the calculation
of the Schwarzschild solution
The Levi-Civita Connection
A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are 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.
Concepts of General Relativity ↑ Riemann Curvature →