# Spacetime

File:Helix.jpg
World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. Note that the orbit of the Earth is (almost) a circle in space, but its worldline is a helix in spacetime.
See Space-time theories of consciousness for the philosophy topic.

In physics, spacetime is a model that combines space and time into a single construct called the space-time continuum. In our universe, this continuum has three dimensions of space and one dimension of time.

Treating space and time on the same footing and as two aspects of a unified whole was devised by Hermann Minkowski shortly after the theory of special relativity was developed by Albert Einstein. This unification is further exemplified by the common practice of expressing time in the same units as space by multiplying time measurements by the speed of light. The concept of spacetime is vital to this theory and also to general relativity, an extension of special relativity, that takes into account gravitation.

Space-times are the arenas in which all physical events take place — for example, the motion of planets around the Sun may be described in a particular type of space-time, or the motion of light around a rotating star may be described in another type of space-time. In any given spacetime, an event is a unique position at a unique time.

## Basic concepts

The basic elements of spacetime are events, these being represented by points in the spacetime. Examples of events include the explosion of a star, or the single beat of a drum.

A spacetime is independent of any observer. However, in describing physical phenomena (which occur at certain moments of time in a given region of space), each observer chooses a convenient coordinate system. Events are specified by four real numbers in any coordinate system.

The worldline of a particle or light beam is the path that this particle or beam takes in the spacetime and represent the history of the particle or beam. Certain types of worldlines - called geodesics (of the spacetime) - are the shortest paths between any two events. Geodesic motion may be thought of as 'pure motion' (inertial motion) in spacetime, that is, free from any external influences.

### Spacetime intervals

The new concept of spacetime brings with it a new concept of distance. Whereas distances are always positive in Euclidean space, the distance between any two events in spacetime (called an interval) may be positive, zero, or negative. The spacetime interval quantifies this new distance (in Cartesian coordinates $\displaystyle x, y, z, t$ ):

$\displaystyle s^2 = \, r^2 - c^2 t^2$

where $\displaystyle c$ is the speed of light, differences of the space and time coordinates of the two events are denoted by $\displaystyle r$ and $\displaystyle t$ , respectively and $\displaystyle r^2 = x^2 + y^2 + z^2$ .

Pairs of events in spacetime may be classified into 3 distinct types based on 'how far' apart they are:

• time-like (more than enough time passes for there to be a cause-effect relationship between the two events; $\displaystyle s^2 < 0$ ).
• light-like (the space between the two events is exactly balanced by the time between the two events; $\displaystyle s^2 = 0$ ).
• space-like (not enough time passes for there to be a cause-effect relationship between the two events; $\displaystyle s^2 > 0$ ).

Events with a positive space-time interval are in each other's future or past, and the value of the interval defines the proper time measured by an observer travelling between them. Events with a spacetime interval of zero are separated by the propagation of a light signal.

## Mathematics of space-times

For physical reasons, a space-time continuum is mathematically defined as a four-dimensional, smooth, connected pseudo-Riemannian manifold together with a smooth, Lorentz metric of signature $\displaystyle \left(3,1\right)$ . The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates $\displaystyle \left(x, y, z, t\right)$ are used.

A reference frame (observer) being identified with one of these coordinate charts, any observer can describe any event $\displaystyle p$ . Another reference frame may be identified by a second coordinate chart about $\displaystyle p$ . Two observers (one in each reference frame) may describe the same event $\displaystyle p$ but obtain different descriptions.

Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing $\displaystyle p$ (representing an observer) and another containing $\displaystyle q$ (another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as 'local observers who can perform measurements in their vicinity' also makes good physical sense, as this is how one actually collects physical data - locally.

For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event $\displaystyle p$ ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples $\displaystyle \left(x, y, z, t\right)$ (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces tensors into relativity, by which all physical quantities are represented.

Geodesics are said to be timelike, null, or spacelike if the tangent vector to one point of the geodesic is of this nature. The paths of particles and light beams in spacetime are represented by timelike and null (light-like) geodesics (respectively).

### Space-time topology

The assumptions contained in the definition of a spacetime are usually justified by the following considerations.

The connectedness assumption serves two main purposes. Firstly, different observers making measurements (represented by coordinate charts) should be able compare their observations on the non-empty intersection of the charts. If the connectedness assumption were dropped, this would not be possible. Secondly, for a manifold, the property of connectedness and path-connectedness are equivalent and one requires the existence of paths (in particular, geodesics) in the spacetime to represent the motion of particles and radiation.

Every spacetime is paracompact. This property, allied with the smoothness of the spacetime, gives rise to a (smooth) linear connection, an important structure in general relativity. Some important theorems on constructing spacetimes from compact and non-pact manifolds include the following:

• A compact manifold can be turned into a space-time continuum if, and only if, its Euler characteristic is 0.
• Any non-compact 4-manifold can be turned into a spacetime.

### Space-time continua and symmetry

For further details, see the article spacetime symmetries

Often in general relativity, space-time continua that have some form of symmetry are studied. Some of the most popular ones include:

## Spacetime in special relativity

The geometry of spacetime in special relativity is described by the Minkowski metric on R4. This spacetime is called Minkowski space. The Minkowksi metric is usually denoted by $\displaystyle \eta$ and can be written as a 4 by 4 matrix:

$\displaystyle \eta_{ab} \, = diag(-1, 1, 1, 1)$

where the Landau-Lifshitz spacelike convention is being used. A basic assumption of relativity is that coordinate transformations must leave spacetime intervals invariant. Intervals are invariant under Lorentz transformations. This invariance property leads to the use of four-vectors (and other tensors) in describing physics.

Strictly speaking one can also consider events in Newtonian physics as a single spacetime. This is Galilean-Newtonian relativity, and the coordinate systems are related by Galilean transformations. However, since these preserve spatial and temporal distances independently, such a space-time can be decomposed into spatial coordinates plus temporal coordinates, which is not possible in the general case.

## Spacetime in general relativity

In general relativity, it is assumed that spacetime is curved by the presence of matter (energy), this curvature being represented by the Riemann tensor. In special relativity, the Riemann tensor is identically zero, and so this concept of 'non-curvedness' is sometimes expressed by the statement: 'Minkowski space is flat'.

Many space-time continua have physical interpretations which most physicists would consider bizarre or unsettling. For example, a compact spacetime has closed timelike curves, which violate our usual ideas of causality. For this reason, mathematical physicists usually consider only restricted subsets of all the possible spacetimes. One way to do this is to study "realistic" solutions of the equations of general relativity. Another way is add some additional "physically reasonable" but still fairly general geometric restrictions, and try to prove interesting things about the resulting spacetimes. The latter approach has led to some important results, most notably the Penrose-Hawking singularity theorems.

## Is space-time quantized?

In general relativity, space-time is assumed to be 'smooth' and 'continuous' (not just in the mathematical sense). In the theory of quantum mechanics, there is an inherent 'discreteness' present in physics. In attempting to reconcile these two theories, it is sometimes postulated that spacetime should be quantized at the very smallest scales. Current theory is focused on the nature of space-time at the Planck scale. Loop quantum gravity, string theory, and black hole thermodynamics all predict a quantized space-time with agreement on the order of magnitude. Loop quantum gravity even makes precise predictions about the geometry of spacetime at the Planck scale.

## Other uses of the word 'spacetime'

Spacetime has taken on meanings different from the 4-dimensional one given above. For example, when drawing a graph of the distance a car has travelled for a certain time, it is natural to draw a 2-dimensional spacetime diagram. As drawing 4-dimensional spacetime diagrams is impossible, physicists often resort to drawing 3-dimensional spacetime diagrams (for example, the Earth orbiting the Sun is a helical shape traced out in the direction of the time axis).

In higher-dimensional theories of physics, for example, string theory, the assumption that our universe has more than four dimensions is frequently made. For example, Kaluza-Klein theory was an attempt to unify the two fundamental forces of gravitation with electromagnetism and used 4 space dimensions with 1 of time. Modern theories use as many as 10 or more spacetime dimensions. These theories are highly speculative, as there has been no experimental evidence to support them.

## History of the concept of space-time

The entire concept was presented by Albert Einstein in 1926 in his article on space-time in the 13th edition of the Encyclopedia Britannica.[1]