Mittag-Leffler summation

Template:Multiple issues Template:Spacetime In physics, spacetime is the setting in which Einstein's theories of relativity are constructed. There are two different definitions of spacetime: Minkowski spacetime from special relativity, and spacetime in general relativity. This article deals with the latter definition. A simple analogy of spacetime is given by the illustration to the right; objects with large masses (like the sun or the planets) cause spacetime to curve and result in the effects of gravity. However, energy also curves spacetime, since matter and energy are the same thing ($E=m{c}^2$).

Definition (mathematical)

Spacetime can be thought of as a differentiable manifold in four dimensions; the Einstein field equations describe curved spacetime by the following

$R_{uv}-\frac{1}{2}{g}_{uv}R+g_{uv}\mathrm{\Lambda }=\frac{8\pi G}{c^4}{T}_{uv}$

Where

• $R_{uv}$ is the Ricci curvature tensor
• $g_{uv}$ is the metric tensor
• $R$ is the scalar curvature
• $\mathrm{\Lambda }$ is the cosmological constant
• $\pi$ is the mathematical constant pi which is approximately equal to 3.14
• $G$ is Newton's gravitational constant
• $c$ is the speed of light
• $T_{uv}$ is the stress-energy tensor

The EFE describe gravity as the result of curved (warped) spacetime; this is the fundamental concept of General relativity. Since spacetime is constructed as a four-dimensional space, it is virtually impossible to fully visualize the real spacetime; that is why physicists and mathematicians use equations and algebraic representations of the geometric ideas for general relativity.

Topology

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The topology of spacetime is very complex and mathematical, but it can be summed up by saying that spacetime (at least from a general relativistic view) is globally and locally smooth, flat and continuous. However, when one factors in quantum fluctuations (of the form $\mathrm{\Delta }E\mathrm{\Delta }t>\frac{h}{4\pi }$) , spacetime is not locally smooth. Such areas as quantum gravity and string theory are trying to resolve this fundamental problem between quantum mechanics and general relativity.

Definition (simple)

Spacetime can be imagined to be the "arena" in which all universal events take place. For example, to specify a point in space, all one needs are a x-coordinate, a y-coordinate, and a z-coordinate; in spacetime, however, one not only needs the x, y, and z coordinates but also the time coordinate.

Within the framework of special relativity, high-speed motion is seen to result in time dilation, which can also occur between two frames of reference as the result of the influence of gravitational fields. Since space-time is a non-discreet (if 4-dimensional) phenomena, when large objects create a curvature within it the resulting curve persists not only through classical 3-dimensional space, but also in time. The larger the gravitational field, the more pronounced the effect of altered passage of time between the frames of reference.

See also

• General relativity
• Spacetime
• Minkowski spacetime
• Time dilation
• The Einstein Field Equations