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Spacetime
Spacetime
In
special relativity and
general relativity,
time
and
three-dimensional space are treated together as a single four-dimensional
manifold called spacetime (alternatively,
space-time; see below). A point in spacetime may be referred to as an
event. Each event has four coordinates (t, x,
y, z).
Just as the x, y, z coordinates of a point depend on the axes
one is using, so distances and time intervals,
invariant in Newtonian physics, may depend on the
reference frame of an observer, in relativistic physics. See
length contraction and
time dilation. This is the central lesson of special relativity.
The central lesson of general relativity is that spacetime
cannot be a fixed background, but is rather a network of evolving relationships.
A spacetime interval between two events is
the frame-invariant quantity analogous to
distance in
Euclidean space. The spacetime interval s along a curve is defined
by
-
ds2 = c2dt2
- dx2 - dy2 - dz2
where c is the speed of light (some people flip the
signs of the equation). A basic assumption of relativity is that coordinate
transformations have to leave intervals invariant. Intervals are
invariant under
Lorentz transformations.
The spacetime intervals on a manifold define a
pseudo-metric called the
Lorentz metric. This metric is very similar to distance in
Euclidean space. However, note that whereas distances are always positive,
intervals may be positive, zero, or negative. Events with a spacetime interval
of zero are separated by the propagation of a
light signal. Events with a positive spacetime 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. Spacetime together with this
pseudo-metric makes up a
pseudo-Riemannian manifold.
One of the simplest interesting examples of a spacetime is R4
with the spacetime interval defined above. This is known as
Minkowski space, and is the usual geometric setting for Special Relativity.
In contrast, General Relativity says that the underlying manifold will not be
flat, if gravity is present, and thus it calls for the use of spacetime rather
than Minkowski space.
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 spacetime can be decomposed unarbitrarily, which
is not possible in the general case.
A
compact manifold can be turned into a spacetime if and only if its
Euler characteristic is 0.
Any non-compact 4-manifold can be turned into a spacetime.
Many spacetimes 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 lead to some important results, most notably
the
Penrose-Hawking singularity theorems.
In mathematical physics it is also usual to restrict the
manifold to be
connected and
Hausdorff. A Hausdorff spacetime is always
paracompact.
Current research is focused on the nature of spacetime at the
Planck scale.
Loop quantum gravity,
string theory, and
black hole thermodynamics all predict a
quantized spacetime with agreement on the order of magnitude. Loop quantum
gravity even makes precise predictions about the geometry of spacetime at the
Planck scale.
Examples of use of spacetime:
-
D. J. Griffiths' Introduction to Electrodynamics
(Upper Saddle River, N. J.: Prentice-Hall, 1989)
-
numerous books with spacetime in title
-
Caltech class: "Spacetime 101"
-
.edu matches online are almost exclusively for
spacetime
Examples of use of space-time:
-
Brehm & Mullin, Introduction to the Structure of Matter
(ISBN
047160531X)
-
Hawking & Ellis, The large-scale structure of
space-time (ISBN
0521099064)
Gravity
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