By Gerard 't Hooft

A presentation of common relativity as a scheme for describing the gravitational box and the equations it obeys. ranging from actual motivations, curved co-ordinates are brought, after which the suggestion of an affine connection box is further. At a later step, the metric box is further.

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Finally, there is room for a cosmological term in the action: √ S = V −g R − 2Λ 1 − 2 (Dµ φ)2 − 12 m2 φ2 . 30) This example with the scalar ﬁeld φ can immediately be extended to other kinds of matter such as other ﬁelds, ﬁelds with further interaction terms (such as λφ4 ), and electromagnetism, and even liquids and free point particles. Every time, all we need is the classical √ action S which we rewrite in a covariant way: Smatter = −g Lmatter , to which we then add the Einstein-Hilbert action: √ S = V −g R − 2Λ + Lmatter .

Exercise: show that this is twice what one would expect if a light ray could be regarded as a non-relativistic particle in a hyperbolic orbit around the star. 40) where a is the major axis of the orbit, ε its excentricity and c the velocity of light. 13. GENERALIZATIONS OF THE SCHWARZSCHILD SOLUTION. a). The Reissner-Nordstrom solution. Spherical symmetry can still be used as a starting point for the construction of a solution of the combined Einstein-Maxwell equations for the ﬁelds surrounding a “planet” with electric charge Q and mass m.

Now however ds = 0. 4), but now we set ds/dτ = 0 , so that Eq. 5) becomes 0 = 1− 2M 2M ˙2 t − 1− r r −1 r˙ 2 − r 2 θ˙ 2 + sin2 θ ϕ˙ 2 . 28) Since now the parameter τ is determined up to an arbitrary multiplicative constant, only the ratio J/E will be relevant. Call this j. Then Eq. 15) becomes u ϕ = ϕ0 + u0 du j −2 − u2 + 2M u3 1 −2 . 29) As the left hand side of Eq. 13) must now be replaced by zero, Eq. 18) becomes u + u = 3M u2 . 31) v + v = 3M A2 cos2 ϕ = v = 3 M A2 1 − 2 1 3 3 M A2 (1 + cos 2ϕ) , 2 cos 2ϕ = M A2 (2 − cos2 ϕ) .