Energy Approximation for Small Velocity - Schutz's "A First Course in GR

[mess1n]

In the Schutz book, "A First Course in General Relativity" (bottom of p.42 if you have it), it states the following:

For small v, the energy is:

E = p^o = m(1-v²)^-(1/2) =(approx) m + (1/2)mv²

I can't figure out why this is! For small v, the LHS will tend to m, and the RHS will tend to m, so it's a valid relation. But how did they derive the RHS from the LHS?

Any help would be appreciated. Let me know if you need some more info.

Cheers,
Andrew

 

[G01]

In the Schutz book, "A First Course in General Relativity" (bottom of p.42 if you have it), it states the following:

For small v, the energy is:

E = p^o = m(1-v²)^-(1/2) =(approx) m + (1/2)mv²

I can't figure out why this is! For small v, the LHS will tend to m, and the RHS will tend to m, so it's a valid relation. But how did they derive the RHS from the LHS?

Any help would be appreciated. Let me know if you need some more info.

Cheers,
Andrew

The right hand side is the Taylor expansion about v=0 of the left hand side. They stop the Taylor expansion at the second term because v is so small that third order and higher terms will be negligible.