By Paul Arthur Schilpp

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Consider next the ratio of circumference to radius as inferred by the inertial observer and the co-rotating observer. By Einstein’s argument, the inertial observer gets the ratio to be √ 2πρ2 ρ−1 . The co-rotating observer obtains 1−v (ρ) the ratio as 2π g(ρ)ρ−1 . Equating the two ratios, determines g(ρ) = ρ2 /(1 − v 2 (ρ)). To determine f (ρ) we use the time elapsed computation. According to the inertial observer a clock at ρ will have tangential speed v(ρ) and will give Examples of Space-Times 23 ∆τρ = ∆τρ=0 1 − v 2 (ρ).

Since the norm of velocity, gµν v µ v ν is constant along a geodesic, the entire geodesic can be labelled as being time-like or null. Thus, both massive and massless, free particles follow time-like (respectively null) geodesics in any space-time. What if a particle has an ‘intrinsic spin’ ? This would be a model for a small gyroscope having spin angular momentum. In special relativity, such a spin will be described by a 4-vector which is space-like. This is because, in the rest frame of a particle (necessarily massive)1 , the √ spin is a 3-vector pointing in some direction and having some magnitude s · s.

24) The particular choice, f (r) = 1 − RS /r, g(r) = f −1 (r) is known as the Schwarzschild space-time. This is not Riemann flat but its Ricci tensor is zero. The Rs is a parameter having dimensions of length and can be related to the mass, M of the spherical body as: RS := 2GM and is known as the c2 Schwarzschild radius of the body. This is completely determined by its mass. On the two-dimensional surfaces defined by t = constant, r = constant, the line element gives to the induced metric which is the standard metric on a sphere.