Relativistic mass effect

Question

Does relativistic mass phenomena only appear while accelerating or even when the object is travelling at constant velocity (say 90% speed of light)?

Answer 1

The formula for relativistic mass is $$ M_r = \frac{M_0}{\sqrt{(1 - v^2 / c^2)}}. $$ where $$ \begin{align} M_r &= \text{relativistic mass}\\ M_0 &= \text{rest mass}\\ v &= \text{velocity}\\ c &= \text{speed of light} \end{align} $$ So the relativistic mass is affected only by the speed.

Answer 2

There is no relativistic mass effect, it came into existence because of inability of an object to gain unlimited speed on applying constant force for unlimited time.

If a constant force $F$ is applied on an object then it causes change in state of motion, momentum. This is good for very short impulse in time but for further duration of time, the present momentum as inertia of an object opposes further change in momentum. Let above force applied for some distance for given time and gain possible maximum speed having momentum $p_0$, then equation of motion is, $$\frac{dp}{dt}+\alpha p=F$$ $$\implies \frac{dp}{dn}=p_0\left(1-\frac{p}{p_0}\right)\tag1$$where $p_0$ is maximum momentum an object can have or say it is momentum gain at first step of impulse where no inertia due to momentum for given force, $dn$ is number of steps or interval of time for force is applied and given by,$$dn=\frac{t}{t_0}=\frac{dn}{dP}\frac{dP}{dE}dE=\frac{dE}{E_0}$$where $t_0$ is time to produce momentum $p_0$ from force $F$, $P$ is rate of energy conversion, $\frac{1}{E_0}$ or $\frac{dn}{dP}$ is minimum amount of energy transferred to produce momentum $p_0$, its thermal equivalent is $\frac{1}{kT}$ and electric equivalent is $\frac{1}{eV}$.

Solving above equation (1) gives,$$p=p_0\left(1-e^{-n}\right)=p_0\left(1-e^{\frac{E}{E_0}}\right)\tag2$$

This equation shows that even apply force for very long time, decline in change in momentum is not due to other losses but entropy which inhibit further energy conversion into work, that is gain in speed.

Now if there is no applied force in (1) and an object have momentum $p_0$ from initial condition, then solution is,$$p=p_0e^{-n}=p_0e^{-\frac{E}{E_0}}\tag3$$

From (1), as $F$ is source term and constant so it forces response to constant after some time and the term containing rate of change in momentum to be zero, that is known as equilibirium condition also, thus $p_0=t_0F$.

The value of maximum momentum from force $F$ can be express into form of,

$p=\int_0^{\infty}\bar np_ndn\tag4$

where $\bar n$ is average number of impulses at $n^{th}$ impulse as there is influence of previous impulses at that point and also there is loss of momentum between two impulses, $p_n=tF=nt_0F=np_0$ is sum at $n^{th}$ impulse. To have expression of $(4)$, we rearrange $(2)$ to have,

$p_0=\int_1^{\infty}p\frac{e^n}{e^n-1}dn\tag*{}$

and multiply this with loss coefficient of $(3)$ and generalize it as $(4)$, so

$\displaystyle p=\int_1^{\infty}\frac{n}{e^n-1}p_ndn=\int_1^{\infty}\frac{n^2F}{e^n-1}dn\tag5$

where, $\bar n=nf(n)=n\frac{1}{e^n-1}$, and $F=p_0$ for $t_0$ having unit value.

Using $(5)$ for energy, replace $F$ by power $P$ instead. Unlike Planck's law of radiation, $(5)$ has variable average value with variable probability function.

Implication of this is that, the value of integral is finite for infinite variable n which is time here. So value of integral solely depends upon value of force, more the force more the speed gained by any object. There is upper limit on speed no matter how long force is applied it can't increase speed infinitely.

Actually in most cases, force is related to square of speed and that is evident in fluid dynamics, terminal velocity or drag. Circular motion clearly shows that acceleration is square of speed at unit distance and opposite to lever function, speed decreases as we go further from centre.

Point is clear that there is relation of density of an object and its ability to gain energy or kinetic energy shows that how mass carry energy in form of speed. But on basis of that to claim that there is relativistic mass because it was thought that on applying force for infinite time not producing infinte speed or infinite gain in energy of an object, there would be mass increase and only significantly at relativistic speed is absurd.