A few key concepts for the brave
But if you want to get the basic idea here without working through the book
above, then just remember that the "spacetime metric" is a thing we
use to measure lengths in spacetime. For example, Archimedes' rule tells us that
the length of the hypotenuse of a right triangle equals the sum of the squares
of two sides. That rule actually defines a space metric called the
"flat" metric or the "Euclidean metric" in two space
dimensions.
Things get a little more complicated when we add time, but not by much.
"Metric" still means "rule for measuring distances", but now
we can measure both space and time.
The subject of Differential Geometry tells how a given spacetime metric is
related to the curvature of that spacetime. Basically, the curvature involves
second-order derivatives of the metric.
The Einstein Theory of General Relativity says that nature is describable by
some spacetime metric,and that nature does not care which set of coordinates we
use for our spacetime metric.
Einstein's Equation says that a combination of curvature tensors is equal to a
tensor representing the distribution of energy in the spacetime from sources
such as particles, radiation, planets, stars, etc.
In other words, matter and energy make spacetime curved. The Einstein Equation
tells us how to calculate the spacetime curvature given the distribution of
energy. In the end we have a set of second-order nonlinear partial differential
equations for the spacetime metric. Once we solve those equations, we know how
to measure distances in our spacetime.
And that's when the surprises begin!
Up to What is the spacetime geometry?
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