A physics-first derivation of the Schwarzschild metric is given. Gravitation is described in terms of the effects of tidal forces (or of spacetime curvature) on the volume of a small ball of test particles (a dust ball), freely falling after all particles were at rest with respect to each other initially. Because this formulation avoids the use of tensors, neither advanced tensor calculus nor sophisticated differential geometry are needed in the calculation. The derivation is not lengthy and it has visual appeal, so it may be useful in teaching.

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I will use the terms Einstein's field equations, Einstein's equation, and just field equations interchangeably. They all mean the same thing.

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Note that in general, the condition of the dust particles in the ball initially being at rest with respect to each other makes sense only for a sufficiently small ball. General relativity does not allow us to assign a physical meaning to relative velocities, hence a relative state of rest, for objects that are not very close to each other.

13.

In standard relativistic Lagrangian mechanics, the action of a free particle between two events is, up to a prefactor, the integral of the line element 12ds. The Lagrangian is then proportional to ds/dt. The Lagrangian (3) is, apart from a prefactor, the square of this, but with the arbitrary time coordinate t replaced by the proper time τ. It can be shown that, if an affine parameter τ, provided by the proper time for massive particles, is chosen as time coordinate in the action integral, extremization of 12(ds/dτ)2dτ yields the same equations of motion as extremization of 12(ds/dt)dt. Therefore, our Lagrangian produces the correct equations of motion, and it is easier to use than the standard Lagrangian, avoiding the appearance of certain square roots. The factor 1/2 has been introduced for convenience, to cancel out some factors of 2, appearing in taking derivatives. Finally, that L is constant is of course due to the fact that L=(1/2)(ds/dτ)2 and that ds2=c2dτ2 for massive particles.

14.

I use this notion that is close in spirit to Einstein's original ideas about gravity, because a number of contemporary authors would object to calling the field experienced by Rindler observers a gravitational one—the spacetime of the Rindler metric is flat.

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The metric is a vacuum solution to Einstein's equation.

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At the beginning of the dust ball's free fall, t̃ may be identified with τ—that is why it is correct to calculate the velocity as x¨(0)τ. As soon as the dust ball center is not coordinate stationary anymore, t̃ and τ become different.

18.

Nevertheless, this is not standard length contraction. The semiaxis of our dust ball in its direction of motion is maximum in its rest frame, i.e., in the frame of the center particle. In fact, δx is not the length of any object, it is the spatial interval between two events (or dust particles) at different Rindler times, the time interval between them being δt. The quantity δxc is the corresponding interval at a fixed proper time in the center particle frame and may therefore be interpreted as an extension of the dust ball at that time.

19.

But see the discussion of initial conditions for δφ̇ in Sec. II D.

20.

The angular frequency as seen by a coordinate stationary observer at r0 will be smaller.

21.

It is, however, not impossible. On presentation of the equation to the computer algebra system MAPLE 17, the latter spat out a general solution dependent on two constants of integration, in an implicit form. Obtaining an explicit form would require the solution of a transcendental equation to invert a function.

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