# On the Nature of ds[in GR]

#### PAllen

A set of differential equations should have a unique solution set corresponding to a given set of boundary conditions. We may try out different techniques--but the aim is to find a solution set that fits into the boundary conditions.If we can do this--the job is done.We can get the correct solution from a set of infinite solutions.
It is easey to see that the boundary conditions do nothing for you. Suppose one solution consistent with them. Do any of uncountably infinite coordinate transformations, you are still consistent with them *and* with your set up.

You really need to let go of the idea of coordinate grid having any meaning (separate from a metric; or unless defined with a fixed operational definition). If you consult books on GR, you will find 100% unanimity that coordinates by themselves are meaningless. More, that points in spacetime have no meaning; only material objects and measurements have meaning.

#### kg4pae

Consider the family of helical paths:
$$\left(ct,R \; cos(\omega t)+R, R \; sin(\omega t), 0\right)$$
Where $$\omega=2\pi/T$$

This helix connects the events (0,0,0,0) and (cT,0,0,0) with a smooth path. Those events are also connected by a straight timelike path.
The only objection I have with this is that the "T" in $$\omega=2\pi/T$$ is the period around the helix, whereas the "T" in (cT,0,0,0) is the time to decay for the muon. Otherwise what you've said is right on.

#### Dale

Mentor
Yes, they are the same. Why do you object to that?

#### kg4pae

Yes, they are the same. Why do you object to that?
Actually I don't object. It's just that it wasn't specified in the original post. Your restriction that the muon exists such that it moves exactly one circle of the helix in its lifetime took some thinking for me to accept, considering that individually muons decay randomly. - But this is just an example of a curved path in spactime, not really about muons, right?

#### PAllen

Actually I don't object. It's just that it wasn't specified in the original post. Your restriction that the muon exists such that it moves exactly one circle of the helix in its lifetime took some thinking for me to accept, considering that individually muons decay randomly. - But this is just an example of a curved path in spactime, not really about muons, right?
Please, the muon was introduced to possibly clarify issues around what types of paths can exist between two events. Nothing else besides one post in this whole thread (before yours) deals with muons. The issue Dalespam was clarifying is long since settled, and even for that, the muon was really irrelevant.

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