Exercise 26 in Schutz's First course in GR

[MathematicalPhysicist]

Homework Statement

The question as follows:

Calculate the energy that is required to accelerate a particle of rest mass ##m\ne 0## from speed ##v## to speed ##v+\delta v## (##\delta v \ll v##).
Show that it would take an infinite amount of energy to accelerate the particle to the speed of light.

Homework Equations

The Attempt at a Solution

Here's what I have done so far, the 4-momentum before is ##(m,mv)## and the 4-momentum after is: ##(E,m(v+\delta v))##, the square of the 4 momentum is conserved, i.e:
$$E^2 - m^2(v+\delta v)^2 = m^2-m^2v^2$$

After rearranging I get the following equation for the energy:

$$E=mv\sqrt{1/v^2+2\delta v /v +(\delta v/v)^2}$$

I think I have a mistake somewhere, since I don't know how to expand this in a Taylor series, I have the expansion ##\sqrt{1+x} \approx 1+1/2 x##, but here I have ##1/v^2## inside the sqrt.

Perhaps I am wrong with the 4-momentum or something else, any tips?

Thanks in advance.

[/B]

 

[MathematicalPhysicist]

I think I got it wrong it should be ##E=\gamma(v+\delta v)mc^2##, and $$\gamma(v+\delta v) \approx 1+1/2 (v+\delta v)^2$$

For ##v +\delta v = c$, we get the E diverges.