QC 5.2: Show Derivative of p w/ Respect to Tau' is 0 for Arbitrary Parameter

[ehrenfest]

Homework Statement

Zwiebach QC 5.2
Tau is the parametrization of a worldline. p is the relativistic momentum
Show that $$\frac{ dp_{\mu}}{d \tau'}} = 0$$implies that $$\frac{ dp_{\mu}}{d \tau'}} = 0$$ holds for an arbitrary paramter $$\tau'(\tau))$$
What needs to be true about the derivative of tau' with respect to tau for tay' to be a good parameter when tau is a good one?

Homework Equations

The Attempt at a Solution

By the chain rule dp/dtau *dtau'/dtau = dp/dtau * dtau/dtau' but we only have tau' as a function of tau and I am not sure whether you can just flip the differentials in that derivative?

 

[George Jones]

By the chain rule,

$$\frac{dp_\mu}{d\tau}=?$$

You need some $\tau'$s on the right.

 

[ehrenfest]

By the chain rule,

$$\frac{dp_\mu}{d\tau}=?$$

You need some $\tau'$s on the right.

$$\frac{dp_\mu}{d\tau}= \frac{dp_\mu}{d\tau'}\frac{d\tau'}{d\tau}$$

But how does that help? We only know that this equals zero. Either term on the right could equal 0. How does this put any restrictions on dtau'/dtau?

 

[George Jones]

How does this put any restrictions on dtau'/dtau?

Imagine Figure 5.2, but with both tau and tau' on it. Also, look at the second paragraph of section 5.2. Can

$$\frac{d\tau'}{d\tau} = 0?$$

 

[ehrenfest]

Imagine Figure 5.2, but with both tau and tau' on it. Also, look at the second paragraph of section 5.2. Can

$$\frac{d\tau'}{d\tau} = 0?$$

No, because then dx/dtau' (where x is a spacetime coordinate on the worldline) is not monotonically increasing. So it must be positive. I see.

So, the second question in this QC should really come before the first because you need the fact that dtau'/dtau is positive in order to prove that dx/dtau' must be zero, right?

 

[Jimmy Snyder]

The first part of the question asks you to show that any $\tau'$ would satisfy equation (5.30) and the second part of the question asks you to remember that only certain $\tau'$ are completely satisfactory. They are almost unrelated questions.

 

[ehrenfest]

I do not see why they are unrelated. To show that 5.30 implies this, we have that

$$(dp_{\mu}/dtau') (dtau'/dtau) = dp_{\mu}/dtau = 0$$

It seems like the second term on the first term on the RHS is only identically zero when the second term on the RHS is always nonzero. This seems to answer both parts of the question but maybe there is something I am missing.