Gravity - Force or Not?

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.

Wikipedia

“Fun” math

Wikipedia

• Einstein field equations
• general relativity
• Riemann tensor
• Ricci tensor
• Einstein tensor

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:

$$\Gamma^i = {\frac{\partial^{2\bar{x}^l}} {\partial x^j\partial x^k}} {{x^i} {\bar{x}^l}}$$

, 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.

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Your Thoughts?

Do you think that a freely falling object will radiate electromagnetic radiation or not? Are we simply all accelerating according to our interaction with matter (Earth in our case)? Until we can devise an experiment that can test this, we will only continue to guess and make assumptions.