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I consider a worldline $x^{\mu}(\lambda)$, where the parameter $\lambda$ parameterizes the world line. Consider now the distance between the two points $x^{\mu}(\lambda+d\lambda)$ and $x^{\mu}(\lambda)$:

$$(x^{\mu}(\lambda+d\lambda)-x^{\mu}(\lambda))^2=\left(\frac{\partial x^\mu}{\partial \lambda}d \lambda\right)= g_{\mu\nu}\frac{\partial x^\mu}{\partial\lambda}\frac{\partial x^\nu}{\partial\lambda} d\lambda^2=g_{\mu\nu}dx^\mu dx^\nu=: ds^2$$.

What is the physical interpretation of the "proper distance" $\sqrt{g_{\mu\nu}dx^\mu dx^\nu}$? Why do we call this a "distance" at all? In flat spacetime we would get $ds=\sqrt{(cdt)^2- dx^2}$ and this is not really a "distance". A physical distance would be for example $dx$!?

Can we encounter physical situations where $x^\mu$ describes for example the movement of a particle and we have $ds^2<0$?

In the case $ds^2<0$ one always defines the proper time. The definition is:

$$c^2d\tau^2=-ds^2$$

What is the physical meaning of this proper time? Is it really the time which an observer who sits always at the point $x^\mu$ would measures in his system? I thought $ds^2$ (and therefore also $d\tau^2$) is an invariant under general coordinate tranformations and therefore the same in every system!?

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    $\begingroup$ "A physical distance would be for example $dx$!" - Unfortunately this is just not true. Our everyday notion of distance does not extend naturally to relativistic setups. The correct (and physical!) notion of distance is actually $ds$. Even under normal circumstances, when you say "the distance from A to B is 1 km" what you really mean is "At their CURRENT locations, the distance from A to B is 1 km" so you are implicitly setting $dt = 0$ in your definition of distance so there $ds=dx$. $\endgroup$ – Prahar May 13 at 6:33
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Despite the fact that spacetime cannot generally be endowed with the structure of an affine space - meaning that we cannot interpret tangent vectors as pointing from one event to another - we can preserve this idea if the events in question are infinitesimally close together. That is, we can interpret $\mathrm d \mathbf x$ as the vector pointing from one event to another which is infinitesimally close by.

The (pseudo) inner product of $\mathrm d\mathbf x$ with itself gives us a scalar quantity which we sometimes call the length, magnitude, or norm of $\mathrm d\mathbf x$ - though of course, since $g_{\mu\nu} \mathrm dx^\mu \mathrm dx^\nu$ can be positive, negative, or zero, this is a generalization of the concept of length with which we are intuitively familiar, and is not truly a norm in the mathematical sense.

This quantity can have several interpretations. To match the convention in your question, we use the signature $(+---)$. If $g_{\mu\nu} \mathrm dx^\mu \mathrm dx^\nu >0$, then the two events are timelike-separated, which means that we can find a frame such that the spatial coordinates of the first event are the same as the spatial coordinates of the second. In other words, there is an inertial observer who is moving in such a way that both events appear to occur in the same place. In the (local) inertial coordinates corresponding to such an observer, $\mathrm d\mathbf x=(c\mathrm dt,0,0,0)$ and so $g_{\mu\nu} \mathrm dx^\mu \mathrm dx^\nu = c^2 \mathrm dt^2$ - the (square of the) time between the two events. As a result, we can interpret $\sqrt{g_{\mu\nu} \mathrm dx^\mu \mathrm dx^\nu}$ - which can be computed in any coordinate system - as the time between the two events as measured by an inertial observer to whom the events appear to occur in the same place.

On the other hand, if $g_{\mu\nu} \mathrm dx^\mu \mathrm dx^\nu<0$, then the two events are spacelike-separated, which means that we cannot find such a frame. However, we can find a frame in which the time coordinates of the two events are the same - that is, the two events are simultaneous. In this frame, $\mathrm d\mathbf x=(0,\mathrm d \vec x)$ and so $\sqrt{-g_{\mu\nu}\mathrm dx^\mu \mathrm dx^\nu} = \sqrt{\mathrm dx^2 + \mathrm dy^2 + \mathrm dz^2}$ is the spatial distance between the two events. Accordingly, $\sqrt{-g_{\mu\nu}\mathrm dx^\mu \mathrm dx^\nu}$ can be interpreted as the spatial distance between two events as measured by an inertial observer to whom those events are simultaneous.

It's important to remember that $g_{\mu\nu}\mathrm dx^\mu \mathrm dx^\nu$ is a generalized notion of distance between events, not between points in space. The physical interpretation of that quantity depends on the nature of the separation of the two events in question.

I thought $\mathrm ds^2$ (and therefore also $\mathrm d\tau^2$) is an invariant under general coordinate tranformations and therefore the same in every system!?

Yes, this is true. $\mathrm d\tau$ is the time between the (timelike-separated) events as measured by an inertial observer to whom the events occur in the same place, and this can be computed in any frame. It coincides with $\mathrm dt$ - the time measured in your frame - if and only if you are one of the aforementioned observers.

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