- #1

- 56

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Let us have a 3D space with non-constant metric. We are in the first region with a euclidian metric.

[itex]ds^2=dx^2+dy^2+dz^2[/itex]

So the distance between two points is got through pythagorean theorem

Then near us we have the second region that has another metric such as

[itex]ds^2=a^2dx^2+b^2dy^2+c^2dz^2[/itex]

where [itex]a,b,c[/itex] - some coefficients

In our region we have line with [itex]A(x_1,y_1,z_1)[/itex] and [itex]B(x_2,y_2,z_2)[/itex] at the ends. So the length of the line is [itex]L_1=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}[/itex]

Then we take our line and put in the second region, so the line ends have coordiantes [itex]C(x_3,y_3,z_3)[/itex] and [itex]D(x_4,y_4,z_4)[/itex]. Its length is

[itex]L_2=\sqrt{a^2(x_3-x_4)^2+b^2(y_3-y_4)^2+c^2(z_3-z_4)^2}[/itex]

The question is:

Will we really observe the line smaller or bigger after putting it into the second region?