Is the Invariant interval $S$ between two points independent of the path taken?

Question

Due to a misunderstanding of what I was asking, I'm re-asking this question (Is the Invariant interval S between the singularity and the present, the same for any point in space in an FLRW universe?) in a much more general sense.

The invariant interval is defined as:

$$ds=\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$$

or rather (as per Hamilton's General Relativity, Black holes, and cosmology equation 2.13):

$$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$$

Because it is a scalar $ds$ may be written as an exact differential form (As per Hamilton equation 2.11 referenced above):

$$ds=\frac{\partial s}{\partial x_{\mu}}dx_{\mu}$$

Where summation over \mu is implied. Note that (for geometric consistency) $\frac{\partial s}{\partial x_{\mu}}$ can be identified with the metric:

$$ds^{2}=\left(\frac{\partial s}{\partial x^{\mu}}dx^{\mu}\right)\left(\frac{\partial s}{\partial x^{\nu}}dx^{\nu}\right)$$

$$=\left(\frac{\partial s}{\partial x^{\mu}}\frac{\partial s}{\partial x^{\nu}}+\frac{\partial s}{\partial x^{\nu}}\frac{\partial s}{\partial x^{\mu}}\right)dx^{\mu}dx^{\nu}$$

Which implies that:

$$\left(\frac{\partial s}{\partial x^{\mu}}\frac{\partial s}{\partial x^{\nu}}+\frac{\partial s}{\partial x^{\nu}}\frac{\partial s}{\partial x^{\mu}}\right)=\left\{ \frac{\partial s}{\partial x^{\mu}},\frac{\partial s}{\partial x^{\nu}}\right\} =g_{\mu\nu}$$

But this expression is familiar, the generalized gamma matrices $\gamma^{\mu}$ are defined by:

$$\left\{ \gamma_{\mu},\gamma_{\nu}\right\} =\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2g_{\mu\nu}$$

Which means that $\frac{\partial s}{\partial x_{\mu}}$ can be identified with the generalized gamma matrices:

$$\frac{\partial s}{\partial x^{\mu}}=\frac{1}{\sqrt{2}}\gamma_{\mu}$$

The second equation can also be written as:

$$ds=\frac{\partial s}{\partial x_{\mu}}dx_{\mu}=\overrightarrow{\nabla}s\cdot d\overrightarrow{r}$$

Integrating now over some arbitrary interval $\{a,b\}$:

$$S=\intop_{a}^{b}\overrightarrow{\nabla}s\cdot d\overrightarrow{r}$$

Via the fundamental theorem of calculus, it is clear that the interval is independent of the path taken between the points $\{a,b\}$. Did I mess this up somewhere?

EDIT: Here's an approach not assuming s is an exact differential (as per the the objections voiced below)

The invariant interval is defined as:

$$ds=\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$$

or rather:

If we wish, this can be rearranged as:

$$0=dx\cdot g\cdot dx-ds^{2}=dx^{\mu}g_{\mu\nu}dx^{\nu}$$

One can write the metric tensor in terms of local basis:

$$g_{\mu\nu}=e_{\mu}\bullet e_{\nu}$$

Where $\cdot$ denotes the standard dot product and $\bullet$ the tensor product. Used in the preceding equation, the above yields:

$$0=\left(e_{\mu}dx^{\mu}\right)\bullet\left(e_{\nu}dx^{\nu}\right)-ds^{2}$$

(note $dx\cdot e=dx^{\mu}e_{\mu}$) This can be simply factored to obtain:

$$0=\left(e_{\mu}dx^{\mu}-ds\right)\bullet\left(e_{\nu}dx^{\nu}+ds\right)$$ Which, for a given metric, gives two different solutions to $ds$.

$$0=\left(e_{\mu}dx^{\mu}-ds\right)\qquad0=\left(e_{\nu}dx^{\nu}+ds\right)$$

Algebraically, this corresponds to the Clifford algebra as we have the relationship:

$$\left\{ e_{\mu},e_{\nu}\right\} =e_{\mu}e_{\nu}+e_{\nu}e_{\mu}=g_{\mu\nu}$$

Which means that $e_{\mu}$ can be identified with the generalized gamma matrice $\gamma_{\mu}$:

$$e_{\mu}=\frac{1}{\sqrt{2}}\gamma_{\mu}$$

Taking either solution for ds individually, the integral for s between two nearby points appears to be independent of the path taken. Note that either solution to $s$ individually could not be considered as proper time between events, but is simply a geometrical invariant.

The relationship between our starting equation and our two solutions now is entirely analogous to that between the Klein gordon equation and the Dirac equation. Solutions to the former are not necessarily solutions to the latter. Apparently no-one liked my first question using differential forms, so I wrote it up this way.

Also, if it eases concerns of undefined intervals, one can simply consider a flat space, since this argument itself is general.

Answer 1

This question does not make sense. The $\mathrm{d}s^2$ notation for the "invariant interval" does not mean that $\mathrm{d}s$ is an actual differential form. The expression $\mathrm{d}s = \frac{\partial s}{\partial x_\mu}\mathrm{d}x^\mu$ is non-sensical because the function $s$ is not defined.

The length of a path $\gamma : [0,1] \to M$ on a pseudo-Riemannian manifold $(M,g)$ is formally defined as $$ l[\gamma] = \int_0^1 \sqrt{g(\dot{\gamma}(t),\dot{\gamma}(t))}\mathrm{d}t$$ where in components the square root is $$ \sqrt{g(\dot{\gamma}(t),\dot{\gamma}(t))} = \sqrt{g_{\mu\nu} \dot{\gamma}(t)^\mu \dot{\gamma}(t)^\nu} $$ and writing $\dot{\gamma}(t)^\mu = \frac{\mathrm{d}x^\mu}{\mathrm{d}t}$ and pulling the $\mathrm{d}t$ into the square root and cancelling it against the $\mathrm{d}t$ would leave one with $\sqrt{g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu}$ if any of these operations were actually mathematically well-defined, but they are not.

The notion of an "invariant interval between points" does not exist in general. You have the lengths of paths as above but it is not clear how one would define the interval between points. The construction that turns a Riemannian manifold into a metric space with distances between points does not work in the pseudo-Riemannian case, as trying to minimize $l[\gamma]$ for all paths between two points can fail, i.e. the infimum over all paths can be $-\infty$, not a meaningful distance.

The object "$\mathrm{d}s = e_\mu\mathrm{d}x^\mu$" exists, but is a rather strange object: The $e_\mu$ are vector fields so this is a vector valued 1-form, and since $g_{\mu\nu} = e_\mu\cdot e_\nu$ but also $g(\partial_\mu,\partial_\nu) = g_{\mu\nu}$ these are simply the coordinate vector fields $\partial_\mu$, i.e. $e_\mu$ has components $(e_\mu)_\nu = \delta_{\mu\nu}$. There is no multiplication of vectors and you cannot assume a priori that $\mathrm{d}s^2$ is actually a square of anything, so the factoring step only seems to work because you are not carefully distinguishing between the dot "product" and actual multiplication.

Since the $e_\mu$ are vector fields, the equation $e_\mu = \gamma_\mu$ for a gamma matrix does not make any sense either, and using an anticommutator doesn't really reflect what's going on either. Using $g(-,-)$ to write the metric instead of as a product, all you have discovered is that (there's a factor of 2 wrong in your equations apart from the abuse of notations) $$ g(e_\mu,e_\nu) + g(e_\nu,e_\mu) = 2g_{\mu\nu},$$ which was the defining property of the $e_\mu$ to begin with.

Finally, this $e_\mu\mathrm{d}x^\mu$ is not an object you can integrate over a path to get a length - since it is a vector-valued one-form, the result of integrating the one-form over a path would still be a vector, not a number.

Answer 2

I agree with ACM's answer; the conclusion is wrong, and there are many counterexamples.

For example, in the $(+---)$ metric, the invariant interval $\int ds$ is equal to the proper time elapsed, for a purely timelike path. But we know that proper time depends on the path taken, for example in the twin paradox, where the moving twin comes back younger.

Answer 3

It's much easier to think about the geometry of the situation.

The 'invariant interval' between two (timelike-separated) events is simply the proper time along a path between the two events. As such it is manifestly path-dependent. As someone else has pointed out, the twin 'paradox' in special relativity is a famous example of the proper time between two events being path-dependent.

Assuming the spacetime is well-behaved (where I'm being vague about what I mean by that because I'm not quite sure I know, but, for instance, no singularities, smooth, no topological oddities at least) then there will always be at least one geodesic between the two events, and the lengths of those geodesics will be local maxima of proper time. In flat spacetime and other 'nice' spacetimes there will be exactly one such geodesic, whose length is a global maximum of proper time.

Answer 4

First of all a remark regarding the question title: the symbol for denoting invariant intervals (in flat spacetime) is "$s^2$". I don't know that in this context there is any terminology for naming "$s$" itself; other than explicitly "the signed square root an interval", as far as $s := \text{sgn}[~s^2~]~\sqrt{s^2~\text{sgn}[~s^2~]}$.

Now, is there some invariant to characterize the geometric relation between two events in more general (at least not necessarily flat) spacetimes (set of events $\mathcal S$)?

Indeed there is, for instance, the "Lorentzian distance $\ell : \mathcal S \times \mathcal S \rightarrow [0, \infty]$".
(Cmp. Beem, Ehrlich, Easley, "Global Lorentzian Geometry".)

Values of Lorentzian distance obey the reverse triange inequality,

$$ \forall \varepsilon_{A B}, \varepsilon_{G H}, \varepsilon_{J K} \in \mathcal S : \ell[~\varepsilon_{A B}, \varepsilon_{G H}~] + \ell[~\varepsilon_{G H}, \varepsilon_{J K}~] \le \ell[~\varepsilon_{A B}, \varepsilon_{J K}~],$$

and they are nonsymmetric ("directional"):

$$ \ell[~\varepsilon_{A B}, \varepsilon_{J K}~] \gt 0 ~~~ \implies ~~~ \ell[~\varepsilon_{J K}, \varepsilon_{A B}~] = 0.$$

Considering a set of events in which participant $A$ took part, $\{ \varepsilon_{A \Xi} \}$, where

$$ \forall \varepsilon_{A P}, \varepsilon_{A Q} \in \{ \varepsilon_{A \Xi} \} : \ell[~\varepsilon_{A P}, \varepsilon_{A Q}~] + \ell[~\varepsilon_{A Q}, \varepsilon_{A P}~] \gt 0,$$

such that the events of set $\{ \varepsilon_{A \Xi} \}$ can be pairwise ordered accordingly, then a value of "length" can be attributed to the ordered set as the infimum of partial sums of Lorentzian distances of any countable (and likewise ordered) subset:

$$L[~\{ \varepsilon_{A \Xi} \}~] := \mathop{\text{infimum}}_{ \large \{ \varepsilon_{A N_k} \} \subset \{ \varepsilon_{A \Xi} \} }\left[ \sum_{j = 0}^{k} \ell[~\varepsilon_{A N_j}, \varepsilon_{A N_{(j + 1)}}~]~\right].$$

If, moreover, set $\{ \varepsilon_{A \Xi} \}$ contains

• an initial event $\varepsilon_{A I}$ such that $\forall \varepsilon_{A P} \in \{ \varepsilon_{A \Xi} \}, \varepsilon_{A P} \not\equiv \varepsilon_{A I} : \ell[~\varepsilon_{A I}, \varepsilon_{A P}~] \gt 0$,

• a final event $\varepsilon_{A F}$ such that $\forall \varepsilon_{A P} \in \{ \varepsilon_{A \Xi} \}, \varepsilon_{A P} \not\equiv \varepsilon_{A F} : \ell[~\varepsilon_{A P}, \varepsilon_{A F}~] \gt 0$,

• and indeed every event $\varepsilon_{A Q}$ of spacetime $\mathcal S$ in which participant $A$ took part inbetween, i.e.
  $(~(\ell[~\varepsilon_{A I}, \varepsilon_{A Q}~] \gt 0) \text{ and } (\ell[~\varepsilon_{A Q}, \varepsilon_{A F}~] \gt 0)~) ~~ \implies ~~ (\varepsilon_{A Q} \in \{ \varepsilon_{A \Xi} \})$,

then the corresponding length of such a set $\{ \varepsilon_{A \Xi} \}$ is $A$'s duration from the indication of having taken part in the initial event, until the indication of having taken part in the final event:

$$\tau A[~{}_I, {}_F~] := L[~\{ \varepsilon_{A \Xi} \}~],$$

and by the reverse triangle inequality follows ("time dilation"):

$$ 0 \lt \tau A[~{}_I, {}_F~] \le \ell[~\varepsilon_{A I}, \varepsilon_{A F}~].$$