How does a particle move under a constant 4-force?

Question

This question pertains specifically to the case of a constant pure 4-force in special relativity, of the form $\textbf{F} = (f_0, f_x, 0, 0)$, where $f_0 = \frac{γ}c \frac{dE}{dt}$. I understand how to solve the case of a constant 3-force in special relativity, but as far as I can tell one cannot use the usual trick of $p(t) = p(0) + f_xt$ since the 3-force is time dependent.

To find a solution, would you need to use the pure force result $\textbf(u \cdot f) = \frac{dE}{dt}$? Any guidance would be helpful!

Answer 1

The confusion here is that many very able physicists have picked up the idea that motion at constant proper acceleration is somehow motion at "constant 4-acceleration" and thus "constant 4-force", but this is simply not true.

For rectilinear motion at constant proper acceleration $a_0 = A^\mu A_\mu$, the invariant size of the 4-acceleration is fixed and so is the direction in space, but the direction in spacetime is not. The direction in spacetime is orthogonal to the 4-velocity and the 4-velocity is certainly changing, and so is the 4-acceleration.

So the problem here is really about clarity and precision of expression. Many people say "constant 4-force" or "constant 4-acceleration" when what they mean is "constant size of 4-force" or "constant size of 4-acceleration".

Compare it to another case: motion in a circle at constant speed in Newtonian physics. Would we say of such motion that the acceleration is constant? We probably would, but we are familiar enough with this case that we don't confuse ourselves: we know the acceleration vector is changing, but its size is constant. Would we say the velocity is constant? No we wouldn't. Would we say the force is constant? We might, or we might not. Strictly, as a vector quantity, it is not constant.

So now let's come back to special relativity. If someone says "calculate the case of motion under a constant 4-force" then really the careful student has little option but to take the statement at face value and try to solve for that case. The solution is difficult and such a force certainly is not pure, and has very little relevance to physics. If the question was "calculate the case of motion under a 4-force of fixed spatial direction and constant invariant size" then the student (whether careful or not) can breath a quick sigh of relief and get on and tackle that standard problem. But if the question said the first while intending to mean the second, then it was an ill-posed question.

(An example of this came up recently in an Oxford physics exam; we need to improve our procedures to catch such things.)

Answer 2

The four-force is the rate of change of four-momentum in respect to the proper time, that is $$ {\bf F} = \frac{d{\rm P}}{d\tau}$$ If it was constant that means that $$ {\bf P}(\tau) = {\bf P}_0 + {\bf F}\tau$$ However that would mean that the mass $$m^2(\tau) = {\bf P}(\tau)\cdot {\bf P}(\tau) = m_0^2 + {\bf P}_0\cdot{\bf F}\tau + {\bf F}\cdot{\bf F}\tau^2 $$ is in general time-dependent. A particle with constant mass can have constant four-acceleration only if ${\bf P}_0\cdot{\bf F}= {\bf F}\cdot{\bf F}$. Assuming $m_0>0$ you can however choose a frame in which ${\bf P}_0=(m_0,0,0,0)$, ${\bf F}=(F_0,\vec{F})$ you have $$ F_0 m_0 = 0 = F_0^2 - \vec{F}\cdot\vec{F}$$ and that means that $F_0=0$, $\vec{F}=\vec 0$, ${\bf F}=0$.

In conclusion, it is impossible to have a constant net four-force on a massive particle with constant mass, except the trivial case of zero net force.

If you allow for mass to change, you have four-velocity $$ {\bf u}(\tau) = \frac{1}{m(\tau)}{\bf P}(\tau) = \frac{{\bf P}_0 + {\bf F}\tau}{\sqrt{m_0^2 + {\bf P}_0\cdot{\bf F}\tau + {\bf F}\cdot{\bf F}\tau^2}}$$ and the four-position $$ (t(\tau),\vec x(\tau)) = {\bf x}(\tau) = \int \frac{{\bf P}_0 + {\bf F}\tau}{\sqrt{m_0^2 + {\bf P}_0\cdot{\bf F}\tau + {\bf F}\cdot{\bf F}\tau^2}} d\tau$$Knowing $t(\tau)$ and $\vec x(\tau)$ you may be able to recover $\vec x(t)$.

Another interesting case is a four-force of constant magnitude but changing space-time direction. An example occurs in the well-known case of a particle accelerating from rest under a constant and uniform 3-force. The result is then hyperbolic motion.

Answer 3

Under John Rennie's guidance; let me know if there's something wrong.

$\textbf{F} = d \textbf{P}/d\tau = m \cdot d \textbf{U}/d\tau$ in the case of a constant force.

$\textbf{F} = m \cdot \textbf{A}$

$\textbf{A} \cdot \textbf{A} = a_0^2$ - > $\textbf{F} \cdot \textbf{F} = \frac{a_0^2}{m^2}$

So we have $\frac{a_0^2}{m^2} = f_x^2 - f_0^2$ for this particular 4-force.

Using the result $sinh(\eta) = \frac{1}{c}\int \:a_0\left(t\right)dt$ yields

$$\frac{\beta}{\sqrt{1-\beta ^2}} = \frac{a_0 t}{c} = \frac{ m \sqrt{f_x^2 - f_0^2} t }{c} $$

which can then be inverted to find the components of the 4-velocity of the particle as a function of time!

Answer 4

The key is to first solve for (4-)momentum, not velocity. You have the basic definition that

$$^④ \mathbf{F} = \frac{d[^④ \mathbf{p}]}{d\tau}$$

where $^④ \mathbf{p}$ is the 4-momentum and $\tau$ is the particle proper time. Hence it's pretty easy to see that

$$[^④ \mathbf{p}](\tau) = {^④\mathbf{p}_0} + [f_0 \mathbf{e}_t + f_x \mathbf{e}_x]\tau$$

where $\mathbf{e}_t$ and $\mathbf{e}_x$ are the unit vectors toward the (coordinate) time and x-directions.

Then note that $[^④ \mathbf{v}](\tau) = \frac{[^④ \mathbf{p}](\tau)}{m}$ and $m = \frac{|^④ \mathbf{p}|}{c^2}$. Note closely - the particle mass will be changing with time viz $m = m(\tau)$! Integration with respect to $\tau$ will give you the worldline, and then you can finally reparameterize to get how it will look in a Lorentz frame.

And yes, that change in mass is very important - when it comes to "ordinary" "constant forces" they actually are not constant 4-forces - they are instead more akin to magnetic force in space-only, in that they cause a change in the direction of the momentum, here the 4-momentum, and not its magnitude. Viz. they are always perpendicular to the 4-momentum and hence also 4-velocity. You can perhaps "emulate" a constant 4-force for a time by piling mass atop an object, but there is no such thing on elementary particles because their masses cannot change - indeed, the electromagnetic interaction when viewed in proper relativistic invariant terms looks like a "giant Ampere's law, 'no Magnetic charge law', and Magnetic Force law" more or less alone but in space-time: in some sense "electro"magnetism is just "space-time magnetism".