Understanding Einstein's Relativity: From Energy Equations to Series Development

[OneEye]

In Relativity, (p. 45), Dr. Einstein says:

...the energy of a material point of mass m is no longer gievn by the well-known expression

$$m { v \over c^2 }$$

but by the expression

$${ m c^2 } \over \sqrt { 1 - { v^2 \over c^2 } }$$

...If we develop the expression for the kinetic energy in the form of a series, we obtain

$$mc^2 + m { v^2 \over 2 } + { 3 \over 8 } m { v^4 \over c^2 } . . . .$$

How does one get from the second equation to the third? What is meant by "develop[ing] the expression... in the form of a series"?

Any help would be more than appreciated.

 

[HallsofIvy]

One way is to use a "Taylor's series" expansion.

If f is any function, analytic at x₀ then
f(x)= f(x₀)+ f '(x₀)(x- x₀)+ (f''(x₀)/2)(x- x₀)²+ (f'''(x₀)/6)(x- x₀)³+ ...
the general term is (f⁽ⁿ⁾(x₀)/n!)(x-x₀)ⁿ where f⁽ⁿ⁾ is the nth derivative.

In particular, use the "McLaurin series" which is the Taylor's series with x₀= 0 and f(x)= (1+ x²/c²)^-1/2, set x= v, and then multiply by mc².

 

[OneEye]

HoI - Thanks - I suspected something like this, but lacked the mathematical toolkit to know what exactly was meant.

Most helpful!