The trivial constant force acceleration math

[valjok]

I accelerate a body by a constant force:
$$a = F/m = F \frac{\sqrt{1-v^2/c^2}}{m_0}$$

I simplify it by fixing F = m0 = c = 1:
$$\frac{dv}{dt} = {\sqrt{1-v^2}$$

This diff equation formalizes the dependence of relativistic body acceleration on its velocity. To get the speed at time t, I solve it rearranging into

$$\int{ \frac{dv}{\sqrt{1-v^2}} = t$$

, which is a handbook integral: t = arcsin v, or v = sin t. This 1) satisfies the equation and, as the Einstein's correction of Newton implies, 2) slows the initially constant acceleration down to zero as v approaches 1 and 3) precludes super-light speeds. However, sine reaches v=1 in finite amount of time while texts tell that we should approach the speed of light asymptotically in t = ∞. Oscillations is not what I expected. Where is the mistake?

 

[Bob S]

When a relativistic particle is accelerated, both the velocity and dynamic (not rest) mass changes. We usually use the momentum p= mv multiplied by a constant, c, to give
pc = mvc = βmc²
So for a constant force,

d(pc)/dt = d(βmc²)/dt = const = d(βγm₀c²)/dt = m₀c² d(βγ)/dt, where m₀c² is the rest mass in energy units.

It is sometimes easier to use the relation

E² = (pc)² + (m₀c²)²
where E is total energy, and E-m₀c² is kinetic energy.

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[DrGreg]

Where is the mistake?

The mistake is that F = ma is not valid in relativity. But F = dp/dt is. See post #16 in "A dark part of special relativity(at least for me)" for details.

 

[valjok]

Thanks, Greg. I understand the mistake now and that Bob tells me how to derive the a. The v(t) solution now turns to out to be non-trivial.