00:03
The two -dimensional 2d space -time span by, it's for part a actually, so i'll write that part a, span by coordinates coordinates v ,x with the line element with the line element will be ds square where ds square is the line element equals minus of x dv square plus 2 dv dx.
01:01
So let this be equation number 1.
01:06
We make a change in coordinates as v dash equals v plus f of x.
01:28
So this implies dv dash equals dv plus lf upon del x into dx.
01:42
So this implies dv dash equals dv plus f dash dx where f dash equals del f upon del x.
02:04
So this implies dv dash square equals dv square plus 2 f dash dx 2 f dash dx dv plus f dash square dx square and this implies dv square equals dv dash square minus 2 f dash dx dv minus f dash square dx square.
02:52
So this is equation number 2.
02:56
Now using equation 2 in 1, we'll get ds square equals minus of x multiplied with dv dash square minus 2 f dash dx dv minus f dash square dx square plus 2 dv dx and this can be written as ds square equals minus of x dv dash square plus x f dash square dx square plus 2 x f dash dx dv plus 2 dv dx.
04:08
So let this be marked as equation number 3.
04:20
2 x f dash dx dv plus 2 dv dx equals 0.
04:33
So this implies x f dash plus 2 equals 0.
04:39
So this implies df equals minus of dx upon x.
04:48
So this implies f equals minus of natural log of x ln x plus c and this is after integration so the integration constant c comes.
05:03
So let this be marked as equation 4.
05:07
Equation 3 implies ds square equals minus of x dv dash square plus x f dash square dx square...