I launched this blog in 1995. Since then, we have published 1603 articles. It's all free and means a lot of work in my spare time. I enjoy sharing knowledge and experiences with you.
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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.
● The geodesic postulate: free falling particles move along geodesics of space-time with the proper time $\tau$ or arc length s as parameter. For particles with zero rest mass (photons), the use of a free parameter is required because for them holds ds=0. From δ∫ds=0 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 xi:
The Riemann tensor is defined as: RναβμTν:=∇α∇βTμ−∇β∇αTμ, where the covariant derivative is given by ∇jai=∂jai+Γjkiak and ∇jai=∂jai−Γijkak.
Here, Γjki=2gil(∂xk∂glj+∂xj∂glk−∂xl∂gjk), for Euclidean spaces this reduces to:
Gαβ:=Rαβ−2gαβR, where
R:=Rαα is the _Ricci scalar_, for which holds: ∇βGαβ=0.
With the variational principle δ∫(≪(gμν)−Rc2/16πκ)∣g∣d4x=0 for variations gμν→gμν+δgμν the Einstein field equations can be derived:
Gαβ=c28πκTαβ , which can also be written as Rαβ=c28πκ(Tαβ−2gαβTμμ)
For empty space this is equivalent to Rαβ=0. The equation Rαβμν=0 has as only solution a flat space.
The Einstein equations are 10 independent equations, which are of second order in gμν. From this, the Laplace equation from Newtonian gravitation can be derived by stating: gμν=ημν+hμν, where ∣h∣≪1. In the stationary case, this results in ∇2h00=8πκϱ/c2.
The most general form of the field equations is:
Rαβ−2gαβR+Λgαβ=c28πκTαβ where Λ is the _cosmological constant_. This constant plays a role in inflationary models of the universe.
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.
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