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## Main Question or Discussion Point

Theorem: In Minkowski space, let ##r(\tau)## define a timelike, ##C^2## world-line W parametrized by proper time for ##\tau## covering the whole real line. Here ##r## is the radius four-vector extending from the origin O to a given point. Then there exists a unique time ##\tau## such that ##r(\tau)## is orthogonal to W at the point defined by ##\tau##.

******* This version is false. See my attempt at revision below. ********

In other words, if you're an observer, every event is simultaneous with you at some uniquely defined time.

How would you go about proving this? I would prefer a proof that avoids invoking any coordinates.

At first I expected this to be trivial, but it seems a little tricky because it seems like you need a topological component to the argument. For example, I had been thinking to argue something like this. Let ##v(\tau)## be the unit tangent vector to W, and let ##f(\tau)=r(\tau)\cdot v(\tau)##. Then it seems obvious, in Minkowski space, that ##f## starts out negative and ends up positive, and since it's continuous it must be zero somewhere. But the fact that ##f## starts negative and ends positive is something that can be false if, for example, you cut out all of Minkowski space for ##t\ge 0##.

I want to avoid making assumptions in the proof that seem reasonable but that require explicit justification to rule out counterexamples such as a flat spacetime with a nonstandard topology.

******* This version is false. See my attempt at revision below. ********

In other words, if you're an observer, every event is simultaneous with you at some uniquely defined time.

How would you go about proving this? I would prefer a proof that avoids invoking any coordinates.

At first I expected this to be trivial, but it seems a little tricky because it seems like you need a topological component to the argument. For example, I had been thinking to argue something like this. Let ##v(\tau)## be the unit tangent vector to W, and let ##f(\tau)=r(\tau)\cdot v(\tau)##. Then it seems obvious, in Minkowski space, that ##f## starts out negative and ends up positive, and since it's continuous it must be zero somewhere. But the fact that ##f## starts negative and ends positive is something that can be false if, for example, you cut out all of Minkowski space for ##t\ge 0##.

I want to avoid making assumptions in the proof that seem reasonable but that require explicit justification to rule out counterexamples such as a flat spacetime with a nonstandard topology.

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