• ## Gravity is an allusion?

The General Theory of Relativity tells us gravity is not a force, gravitational fields don’t exist. Objects tend to move on straight paths through curved spacetime.

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### “Fun” math

Wikipedia

The basic principles of general relativity are:

● The geodesic postulate: free falling particles move along geodesics of space-time with the proper time or arc length as parameter. For particles with zero rest mass (photons), the use of a free parameter is required because for them holds . From the equations of motion can be derived:

● The principle of equivalence: inertial mass gravitational mass gravitation is equivalent with a curved space-time were particles move along geodesics.
● By a proper choice of the coordinate system, it is possible to make the metric locally flat in each point :

The Riemann tensor is defined as: , where the covariant derivative is given by and 6 .

Here, , for Euclidean spaces this reduces to:

, are the Christoffel symbols.
For a second-order tensor holds:
,
,
and
.

The following holds: .

The Ricci tensor is a contraction of the Riemann tensor:
,

which is symmetric:
.

The Bianchi identities are:
.

The Einstein tensor is given by:
, where
is the Ricci scalar, for which holds: .
With the variational principle
for variations the Einstein field equations can be derived:

, which can also be written as

For empty space this is equivalent to . The equation has as only solution a flat space.

The Einstein equations are 10 independent equations, which are of second order in . From this, the Laplace equation from Newtonian gravitation can be derived by stating: , where . In the stationary case, this results in .

The most general form of the field equations is:
where is the cosmological constant. This constant plays a role in inflationary models of the universe.