# What is spacetime 3

## Spacetime

Spacetime *space-time*) is an artificial word that appropriately expresses an essential result of the theory of relativity: space and time are no longer dimensions that are independent of each other. As the Lorentz transformation of the special theory of relativity (SRT) shows, space and time form a continuum! The time and the three space coordinates are closely interwoven and form one **four-dimensional spacetime**.

### Inventor of the term *Spacetime*

The term *Spacetime* was founded in 1908 by mathematician *Hermann Minkowski* (1864 - 1909) who invented a new representation of the theory of relativity using vectors of four. This elegant formulation is still in use today. One funny anecdote is that Minkowski *Albert Einstein* Was a mathematics teacher at the Zurich Polytechnic. Despite his talent, Einstein did not necessarily excel in Minkowski's classes. Minkowski was all the more astonished when he learned that this Einstein had invented the theory of relativity. So he responded with the following comment:

'I really wouldn't have believed Einstein to do that.'

(Source: biography **Albert Einstein** of *Thomas Bührke*, dtv 2004)

### dynamic and curved spacetime

Space and time no longer exist as absolute quantities as in classical physics, but are themselves **dynamic object**, physical quantity, a complex one **Tensor field**, a metric. This spacetime can occur in the absence of matter and energy*flat* then it is the Minkowski metric. However, space-time is generally curved by masses or, more generally, forms of energy. Then the general theory of relativity (GTR) has to be used to describe it. The curvature of spacetime is particularly pronounced in black holes, which are described by the Schwarzschild metric (static) or Kerr metric (rotating). The curvature only becomes particularly high close to the black hole and disappears at large distances. The relativists say: there the metric is asymptotically flat.

It is by no means a trivial task how one has to imagine space-time. In principle we are used to a four-dimensional world: In a room we fill three dimensions of space, namely length, width and height - one more, the other less. At a later point in time, we may find ourselves in a different place in the room: we have moved through time. But how do these four dimensions form a continuum? To illustrate this, you have to use a few tricks and e.g. suppress a space dimension and the time dimensions. A fairly simple idea of spacetime is then an analogy to a stretchable one **Rubber skin**. In this simple 2D model, the information about the time dimension and one spatial dimension is lost. A rubber skin stretched in a frame forms a flat surface (= flat space-time without the presence of masses). If you put a mass, e.g. a heavy metal ball on the rubber skin, it will be curved. You get a 'dent' in spacetime, a curved spacetime (see figure above right). The depth of the dent is a measure of the curvature of space-time.

The dent is a result of an extended mass. You can continue this game and let the mass collapse in gravitational collapse. The result is a black hole. Illustrative representations of this are diagrams from **Gravity funnels**as presented in the entry lapse function. The funnel does not close because there is a singularity of curvature here.

### mathematical representation of spacetime

Mathematically, spacetime becomes unambiguous through the **metric tensor** (Metric) or alternatively that **Line element****ds ^{2}** described. The components of the metric tensor are in the line element as coefficients. Because the metric tensor is a second order tensor, it is 4 × 4 = 16 coefficients. Such a tensor can be written down as a matrix with four columns and four rows. The symmetry of the tensor reduces the 16 to only 4 + 3 + 2 + 1 = 10 independent components. Now it depends on the complexity and the symmetry properties (see also isometries) of the respective spacetime how many of the ten independent coefficients are different from zero. In the case of high symmetries, such as in the Minkowski geometry or the Schwarzschild geometry, the metric tensor is diagonal: then there are only four independent coefficients other than zero on the matrix diagonal - the remaining entries are zero.

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