Help me with this equation from Invariance of interval

dpa

Here is an equation from proof of invariance of interval:
This equation is from bernard schutz's first course in GR:



I could not understand what M stands for.

Can someone help me with this?

I donot have advanced knowledge. I am a beginner UG.

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DaleSpam

M is the metric, in most other books it is labeled g. It is essentially the object which maps changes in the coordinates (Δx) to distances and times (Δs).

For example, if you have a spherical coordinate system in flat spacetime then your coordinates would be [itex]x=(t,r,\theta,\phi)[/itex], but if θ changes by 1, how much does s change? That is what M contains.

PSMD

To Piggy Back onto this question, I am confused as to the Note regarding this metric. Why exactly may we assume that the element Mab=Mba? I do not understand what they mean by the fact that Mab+Mba only appears when b doest not equal a.

Thank You,

-PD

Nugatory

The equation you've quoted above is only one particular application of the metric. More generally, the metric is used to calculate the dot-product of two vectors [itex]X[/itex] and [itex]Y[/itex] with components [itex]X^{i}[/itex] and [itex]Y^{i}[/itex]: [itex]X\cdot{Y} = g_{ab}X^{a}Y^{b}[/itex]
(Here I've written the metric as a lower-case g instead of an upper-case M because that's more common, and I've used the Einstein summation convention in which we sum across pairs of repeated indices - you'll see this a lot in GR)

You'll notice that if you compute the dot-product of a vector with itself, you'll get exactly the equation you quoted (with the [itex]\Delta{\chi}^{a}[/itex] being the components of the vector and the dot-product being the square of the length of the vector, that is, the interval between the two ends of the vector).

Because the dot-product is commutative [itex]X\cdot{Y}=Y\cdot{X}[/itex], it follows that [itex]g_{ab}=g_{ba}[/itex]. When a tensor has this property, we say that it is symmetric, and because the metric tensor is defined to produce the dot-product of vectors, it has to be symmetric.