User:Stigmatella aurantiaca/sandbox

$\gamma =1/{\sqrt {1-v^{2}/c^{2}}}$

$10^{-18}<\partial a/a<10^{-13}$

$\Delta f/f=4.4647\times 10^{-10}$

In his novel The Time Machine (1895), H. G. Wells wrote, "There is no difference between time and any of the three dimensions of space except that our consciousness moves along it". Furthermore, "any real body must have extension in four directions: it must have Length, Breadth, Thickness, and Duration".

Wells described a four-dimensional spacetime where time and space are on a precisely equal footing. If we let c be the proportionality constant used to convert seconds into meters, the distance between two points can be described using the four dimensional version of the Pythagorean theorem:

$(\Delta d)^{2}=(c\Delta t)^{2}+(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}$

In the above, the proportionality constant c has units of speed. To a mathematician, the above metric (distance function) precisely describes the geometric properties of Wells' version of spacetime. In particular, given that time and space are on a precisely equal footing, the hero of his story could, with mechanical assistance, travel freely back and forth through time in exactly the same manner in which he could walk to and fro on the surface of the Earth. This, of course, is the essential plot device underlying Wells' novel.

The geometric properties of the real world are significantly different than the imaginary world that Wells envisioned. Experiment and observation informs us that although time comes in as a fourth dimension, it needs to be treated differently than the spatial dimensions. Instead of distance $\Delta d$ , the "spacetime interval" $\Delta s$ is described by the following metric:

$(\Delta s)^{2}=-(c\Delta t)^{2}+(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}$

The change of sign results in a number of interesting properties. For example, an upper speed limit exists whose value is equal to c. Unlike the situation in Wells' fantasy universe, it is impossible to travel backwards in time.

In 1909, in his famous Space and Time lecture, Hermann Minkowski demonstrated that from the above definition of the spacetime interval, it was possible to derive the entirety of special relativity.