Deriving Energy in Special Relativity: The Principle of Extremal Aging

[H-bar None]

I'm reading Taylor and Wheeler's, Exploring Black Holes.

I was doing okay until I reached their derivation of energy in Special Relativity.

They arrived at this equation:

$$\frac{t}{\tau} = \frac{E}{m}$$

Tau is proper time, t is the frame time, E is energy and m is mass.

The authors used the Principle of Extremal Aging to derive the equation. How did they arrive at E/m as a constant of motion?

 

[robphy]

They arrived at this equation:

$$\frac{t}{\tau} = \frac{E}{m}$$

Tau is proper time, t is the frame time, E is energy and m is mass.

t is the time-component of the position 4-vector with magnitude $$\tau$$.
$$t=\gamma \tau$$

E is the time-component of the momentum 4-vector with magnitude $$m$$.
$$E=\gamma m$$

 

[H-bar None]

I sort of understand the 4-vector part. How does that relate to "E/m"?
I'm going to do some more reading check back with you later on in life.

Could go into a litte more detail, maybe I'm missing something.
Thanks for the response.

 

[robphy]

Since $$t=\gamma \tau$$, we have $$\frac{t}{\tau}=\gamma$$.
Since $$E=\gamma m$$, we have $$\frac{E}{m}=\gamma$$.

Thus, $$\frac{t}{\tau}=\gamma=\frac{E}{m}$$.