When giving an explicit formula for a metric in terms of a coordinate chart, e.g. the Schwarzschild metric, it is conventional to use the notation [itex]ds^2 = g_{\alpha\beta} dx^\alpha dx^\beta[/itex] (with the right hand side explicitly written out).(adsbygoogle = window.adsbygoogle || []).push({});

This notation has an unfortunate side-effect. If you take it literally, thens, evaluated along a spacetime curve, can be either real or imaginary, depending on whether the curve is spacelike or timelike and what your metric signature is. This could be potentially confusing to students of the subject.

Are you aware of any alternative notation that authors have used to get around this notational ugliness?

You might be tempted to write something likedQinstead ofds^{2}, and then consider separately [itex]ds = \sqrt{dQ}[/itex] and [itex]d\tau = \sqrt{-dQ/c^2}[/itex]. However that notation is not "infinitesimally correct" as you can't equate a squared infinitesimal with a single infinitesimal.

So are there any neater solutions?

This question arose because of this comment in another thread:

kev said: ↑I was not being critical of your notation and I realise that you are just using the conventional notation as used in text books. It is just that ds is indeed a grey area and can mean many different things. It can be seen from the equations you listed that ds can have different units depending on the context and that can be confusing for beginners. Because of the many faces of ds it can even appear to ignore the rules of algebra because when switching from the +--- to -+++ signature the sign of ds does not change which is only true in algebra if ds has the value zero. DrGreg said: ↑I was a bit lax in my notation, and this is indeed a bit of a notational grey area.

To keep things simple, let's just consider flat 2-d spacetime (i.e. 1 space + 1 time coord, with no gravity). Different authors will tell you that the metric is given by one of the following equations.

[tex] ds^2 = c^2 dt^2 - dx^2 [/tex] (I)

[tex] ds^2 = dt^2 - dx^2 / c^2 [/tex] (II)

[tex] ds^2 = dx^2 - c^2 dt^2 [/tex] (III)

[tex] ds^2 = dx^2 / c^2 - dt^2 [/tex] (IV)

(I) and (II) are referred to as a (+---) metric signature, while (III) and (IV) are referred to as a (-+++) signature. When you get more experienced in the subject, you might even mentally switch from one definition to another without explicitly saying so.

IMHO ds should be explicitly stated in terms of other variables to avoid confusion. For example your list could be restated as:

[tex] c^2 d\tau^2 = c^2 dt^2 - dx^2 [/tex] (I)

[tex] d\tau^2 = dt^2 - dx^2 / c^2 [/tex] (II)

[tex] -(c^2 d\tau^2) = dx^2 - c^2 dt^2 [/tex] (III)

[tex] -d\tau^2 = dx^2 / c^2 - dt^2 [/tex] (IV)

That way it becomes clear that expressions (I) and (III) have units of proper distance and expressions (II) and (IV) have units of proper time. It also becomes clear that the rules of algebra still apply and that changing the metric signature of the right hand side also changes the sign of the left hand side (ds).

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# Ds notation for the metric

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