Relativistic Centripetal Acceleration

The standard Newtonian centripetal acceleration is:

$$g = \frac{V^2}{R}$$

where \(V\) is the rectilinear velocity being bent into a circular motion and \(R\) is the radius of the circular trajectory that it is being bent into.

When the velocity is closer to lightspeed, the Lorentzian gamma-factor gets involved:

$$\gamma^2 = \frac{1}{1-\beta^2}$$

where \(\beta\) is the ratio of the velocity to the speed of light, \(\frac{V}{c}\).

The centripetal acceleration – as felt by the observer in circular motion – becomes:

$$g = \frac{\gamma^2V^2}{R} = \frac{c^2(\gamma^2-1)}{R}$$

To get a feel for the numbers, at what velocity is the force per unit mass equivalent to the Newtonian case of the speed of light? In otherwords…

$$g = \frac{c^2(\gamma^2-1)}{R} = \frac{c^2}{R}$$

i.e. \(\gamma^2-1 = 1\) or \(\gamma^2 = 2\), thus \(\gamma = 1.4142…\)

To convert from \(\gamma\) to \(\beta\) requires some hyperbolic functions:

$$\beta = tanh(acosh(\gamma))$$

…which, in this case of \(\gamma = 1.4142\) means \(\beta = 0.70710\).

In Stephen Baxter’s novel “Ring” the starship “The Great Northern” sets out in a huge loop to take it 5 million years into the future. It accelerates at 1 gee for the whole journey, for the sake of its human passengers, and its total trip time is about 1,000 years, meaning it must have a gamma-factor of about 5,000. Such a gamma-factor would require a loop of roughly 25 million light-years radius, as the above equation implies, thus the journey would be much longer than 5 million years.

Reference

Yongwan Gim, Hwajin Um, Wontae Kim, “Unruh temperatures in circular and drifted Rindler motions”, (Submitted on 28 Jun 2018) https://arxiv.org/abs/1806.11439 [accessed 02 August 2018]

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