6.1. Lorentz-factor (dt/dτ) from the metric

Here we outline the basic relationships using physical units for the benefit of folks interested in some concrete calculations.

The metric equations we've discussed here are the Minkowski metric, the Schwarzschild metric, and the flat-space radar-time constant proper-acceleration metric. The Lorentz factor defined as dt/dτ for these is simply a rearrangement of the metric equation itself: Divide through by cδτ, and solve for δt/δτ by moving terms around and taking the square root.

The metric equations discussed here are the Minkowski metric, the Schwarzschild metric, and the flat-space radar-time constant proper-acceleration metric. The Lorentz factor defined as dt/dτ for these is simply a rearrangement of the metric equation itself: Divide through by cδτ, and solve for δt/δτ by moving terms around and taking the square root.

For the (1+1)D Minkowski (flat-space) metric

this yields the special-relativistic Lorentz factor γ ≡ dt/dτ = Sqrt[1+(w/c)2] = 1/Sqrt[1-(v/c)2] in terms of proper-velocity w ≡ dx/dτ or coordinate-velocity v ≡ dx/dt.

For the (1+1)D Schwarzschild (gravitational) metric with only radial motion, we start with:

.

On earth's surface the metric equation doesn't differ much from the Minkowski case since rs/rfar ≈ 1.39117×10-9, given that event-horizon radius rs = 2 GM/c2 ≈ 8.87 [mm], where G is the universal gravitation constant and M the earth's mass. Nonetheless, most of us will agree before walking off the edge of a cliff that the effects of gravity may be far from negligible! 

The Lorentz-factor in this case becomes the product of a motion-related and a radius-dependent term, namely γ ≡ dtfar/dτ = Sqrt[1+(w/c)2]/Sqrt[1-rs/rfar], where bookkeeper coordinates tfar and rfar represent ``far-coordinates" i.e. values determined using the synchronized clocks and yardsticks of observers in flat space far away from our gravitational object. 

Finally the (1+1)D accelerated-frame radar-time metric, by comparison, looks like

.

We will only be considering objects which are a fixed radar-time distance xα from our accelerated traveler (and hence in this specific sense ``stationary" to our traveler), the Lorentz factor of interest is γ ≡ Δtα/Δτ = e-2xαα/c2.