Why Is the Formula for Relativistic Mass Wrong?
And What Is the Right One?
Many have heard of formula m = m₀/√(1–v²/c²), which showing that mass m of an accelerated object increases to infinity when the object’s velocity v approaches the speed of light c, as 1–v²/c² in the denominator comes close to zero, and division by zero gives infinity. Here m₀ denotes the rest mass or initial mass of the object when it does not move. Where does this denominator come from?
That is rate of time dilation: time in fast moving objects slows down by factor D. That was Einstein’s first discovery.
Now we can simplify mass formula to m = m₀×D. But there is a small mistake in this formula: it should be corrected to m = m₀×D²! Why is that?
Mass measures inertia (the response of an object to a force applied). This response is observed through the change in the object velocity, which is called acceleration. Newton’s 2nd law of motion F = m×a for a force F acting on an object of mass m. That force F causes a change in velocity of the object, and that is acceleration a. Heavier objects react less than lighter objects when the same force is applied to them. If mass increases by a factor x, then acceleration decreases by the same factor — that is the meaning of Newton’s 2nd law: we can say that mass and acceleration are inversely related. Now, what happens when time slows down by a factor D? Let me quote Universal Inverse-Square Laws without Singularities!
For an object moving at speed v m/sec in a slower-time area, let’s denote such speed as v(D), where D is time dilation in this area, compared to another area where we stay and look at this moving object. What is the velocity of this object from our time perspective? In a second of time, in the slower-time area object moves v(D) meters, but in our (faster) time area, our clocks ticked not once, but D times (D seconds passed for us), thus velocity that we observe is v(D)/D (the same distance divided by our time passed). Let’s denote such observed velocity as v(1) (meaning v(for D=1), since there is no time dilation for us). We have v(1) = v(D)/D. Apparently, it is the mere consequence of our clocks ticking more often, thus, any object moves a shorter distance at 1 tick of the faster-ticking-clock. Velocity is measured in m/sec, and since velocity is just change in position per second, with the whole change in position being the same in our time and slower time, but in more seconds for us, change in position per our second is smaller than it is in slower time (for slower-ticking clock). Let’s go further: consider change in velocity per second — that is acceleration, it is denoted by a m/sec². Again, because time interval at which we track change in velocity is D times shorter in our time, and velocities are already D times smaller in our time, then change in velocities in our time is D×D=D² times smaller than in slower time: a(1) = a(D)/D².
a(1) is acceleration observed from perspective of a faster time than the object’s time, which is D times slower than the observer’s time. Observed acceleration decreases by factor D². Thus, mass as measurement of inertia (resistance to force) increases by factor D² (mass m is inversely related to acceleration a, as we discussed right above the quote). We just proved m = m₀×D².
P.S. Time dilation does not cause any distance changes, as widely assumed. It only causes refraction by Snell’s law with refractive index D. Read chapter 1 in Time Matters (the free eBook also available on Amazon). And check 2-page story
P.P.S. If interested in how that changes energy of an object by half, continue to
