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I am going through my notes trying to get things straight in my head and something is confusing me. I also have the feeling it will turn out not to be very complicated, which is why I am a bit frustrated with this.

I know that there is something used in general relativity called the Lagrangian, which is given by [tex] L = g_{\nu\mu}\frac{dx^\nu}{d\tau}\frac{dx^\mu}{d\tau}[/tex], where the dx^{[tex]\mu[/tex]}and dx^{[tex]\nu[/tex]}refer to the coordinates of the chosen coordinate system, and g is the metric, where the subscript refer to elements of the matrix it can be written as.

I also know there is something called the spacetime interval ds. As I understand things, in flat space, we use Cartesian coordinates and we have a metric, which (I think) is called the Minkowski metric:

[-1 0 0 0]

[ 0 1 0 0]

[ 0 0 1 0]

[ 0 0 0 1] and the metric elements and the interval are related by:

[tex] ds^2 = g_\nu_\mu dx^\nu dx^\mu [/tex], where [tex] \mu,\nu = 0,1,2,3 [/tex].

As I understand things, if the spacetime involved was not flat, then these diagonal elements would be different, right?

Then we say that [tex] dx^0 = cdt, dx^1 = dx, dx^2 = dy, & dx^3 = dz [/tex], and noting that if [tex] \nu \neq \mu [/tex] then the corresponding metric coefficients are zero.

this gives us, using Einstein's summation convention, that:

[tex] ds^2 = g_0_0(cdt)(cdt) + g_1_1(dx)(dx) + g_2_2(dy)(dy) + g_3_3(dz)(dz) [/tex]

[tex] = -c^2dt^2 + dx^2 + dy^2 + dz^2 [/tex].

I understand this as being the separation between 2 events separated by position or time or both.

Then looking back at the Lagrangian equation above, it gives this:

[tex] L = -c^2(\frac{dt}{d\tau})^2 + (\frac{dx}{d\tau})^2 + (\frac{dy}{d\tau})^2 + (\frac{dz}{d\tau})^2 [/tex]

My first question is, what is the interpretation of the Lagrangian exactly? All I have seen in my notes is that it gets stuck into the Euler-Lagrange equation, and sometimes conserved quantities or equations of motion pop out. I remember using the Lagrangian L=T-V in classical mechanics, and I do not see the relationship between the two (nor did I ever really understand what the significance of the difference between kinetic and potential energy was).

Does it really have a physical meaning or is it just some weird mathematical trick I have to blindly use to make equations of motion appear?

The other thing that was making me confused was a part in my notes where the interval/Schwarzschild metric for a particle on a radial trajectory towards a spherical mass was given, using spherical polar coordinates, as:

[tex] ds^2 = -c^2dt^2(1 - \frac{2GM}{rc^2}) + \frac{dr^2}{1-\frac{2GM}{rc^2}} [/tex].

Apparently I am supposed to be able to find a Lagrangian for the particle from this interval, but I don't get what I am supposed to do with it to find this. I am not really even comfortable with the way "interval" and "metric" seem to be interchangeable.

Why then, if the interval is given by [tex] ds^2 = g_\nu_\mu dx^\nu dx^\mu [/tex], is the Lagrangian given by [tex] L = g_{\nu\mu}\frac{dx^\nu}{d\tau}\frac{dx^\mu}{d\tau}[/tex]? These two things look similar, i mean, there is only a derivative with respect to tau to tell between them, but its clearly not as simple a thing as, say, [tex] L = \frac{ds}{d\tau} [/tex].

Anyone able to set me straight here?

Thanks.

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# Struggling with the relationship between the Lagrangian L and the interval ds?

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