Consider a 3D space whose line element is given by: ds^2 =...

Consider a 3D space whose line element is given by: $ds^2 = \frac{dr^2}{1 - 2\mu/r} + r^2(d\theta^2 + \sin^2\theta d\phi^2)$ and determine the following quantities: a. the radial distance between the sphere of radius $r = 2\mu$ and the sphere of radius $r = 3\mu$ b. the 3-volume contained between the two spheres of item a

Transcript

00:01 So we are given with three spheres at o, p and q with the given coordinates of unit radius.

00:07 And another sphere of radius unit value is placed in the red diagram or red sphere as shown in the diagram.

00:14 And we need to find the center r of this sphere where this is o, this sphere.

00:22 And since we are taking this as positive, so this will be our q and this will be our p sphere.

00:31 Need to also find the vector rop r .oq, r .pq, r .p .r .p .r .r .p .r.

00:38 Where i .j represent vector from center i to center j.

00:42 Okay.

00:43 Now if we look this arrangement from the top, that is in yx plane.

00:47 So let's have our y plane, sorry, y -axis and our x -axis.

00:51 So we have one sphere here.

00:53 That will be as a circle.

00:56 Then another one is here and one is here.

01:00 Okay.

01:01 Now if we join this, centers of this will form equilateral triangles since all have unit 1.

01:06 Now this is center with x as 0, y is 0 and z as 0.

01:13 Now this has center x as root 3, y as plus 1 and z s for this at this plane it is 0.

01:26 And similarly here we'll have root 3 minus 1 and 0.

01:29 Now to find the center of this, so we can find the centroid of this triangle, right, and that will be nothing but the two coordinates of this sphere sitting on the top of these three spheres, right? so we can easily calculate the centroid of this.

01:45 Centroid will be, let's take it rx and ry, since we need this.

01:53 So rx, ry will be addition of all three x coordinate...

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