Minkowski spacetime

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Example of a light cone.

In special relativity, the Minkowski spacetime is a four-dimensional manifold, created by Hermann Minkowski. It has four dimensions: three dimensions of space (x, y, z) and one dimension of time. Minkowski spacetime has a metric signature of (-+++) and is always flat. The convention in this article is to call Minkowski spacetime simply spacetime. (It should be noted, however, that Minkowski spacetime is only applicable in special relativity; general relativity used the notion of curved spacetime to describe the effects of gravity and accelerated motion).

Definition(s)[change | change source]

Mathematical[change | change source]

Spacetime can be thought of as a four-dimensional coordinate system in which the axes are given by

(x, y, z, ct)

They can also be denoted by

(x^1, x^2, x^3, x^4)

Where x^4 represents ct. The reason for measuring time in units of the speed of light times the time coordinate is so that the units for time are the same as the units for space. Spacetime has the differential for arc length given by

ds^2=-c^2dt^2+dx^2+dy^2+dz^3

This implies that spacetime has a metric tensor given by

g_{uv}=\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}

As before stated, spacetime is flat everywhere; to some extent, it can be thought of as a plane.

Simple[change | change source]

Spacetime can be thought of as the "arena" in which all of the events in the universe take place. All that one needs to specify a point in spacetime is a certain time and a typical spacial orientation. It is hard (virtually impossible) to visualize four dimensions, but some analogy can be made, using the method below.

Spacetime diagrams[change | change source]

In the theory of relativity both observers assign the event at A to different times.

Hermann Minkowski introduced a certain method for graphing coordinate systems in Minkowski spacetime. As seen to the right, different coordinate systems will disagree on an objects spacial orientation and/or position in time. As you can see from the diagram, there is only one spacial axis (the x-axis) and one time axis (the ct-axis). If need be, one can introduce an extra spacial dimension, (the y-axis); unfortunately, this is the limit to the number of dimensions: graphing in four dimensions is impossible. The rule for graphing in Minkowski spacetime goes as follows:

1) The angle between the x-axis and the x'-axis is given by tan \theta=\frac{v}{c} where v is the velocity of the object

2) The speed of light through spacetime always makes an angle of 45 degrees with either axis.

Spacetime in general relativity[change | change source]

In the general theory of relativity, Einstein used the equation

R_{uv}-\frac{1}{2}g_{uv}R=8 \pi  T_{uv}

To allow for spacetime to actually curve; the resulting effects are those of gravity.

Related pages[change | change source]