The density of a stationary body is equal to `rho_0`. Find the velocity (relative to the body) of the reference frame in which the density of the body is `eta=25%` greater than `rho_0`.

The density of a stationary body is equal to `rho_0`. Find the velocit

1.	The density of a stationary body is equal to `rho_0`. Find the velocity (relative to the body) of the reference frame in which the density of the body is `eta=25%` greater than `rho_0`.
Answer» We define the density `rho` in the frame K in such a way that `rhodxdydz` is the rest mass `dm_0` of the element. That is `rhodxdydz=rho_0dx_0dy_0dz_0`, where `rho_0` is the proper density `dx_0`, `dy_0`, `dz_0` are the dimensions of the element in the rest frame `K_0`. Now `dy=dy_0`, `dz=dz_0`, `dx=dx_0sqrt(1-v^2/c^2)` if the frame K is moving with velocity, v relative to the frame `K_0`. Thus `rho=(rho_0)/(sqrt(1-v^2/c^2))` Defining `eta` by `rho=rho_0(1+eta)` We get `1+eta=(1)/(sqrt(1-v^2/c^2))` or , `v^2/c^2=1-(1)/((1+eta)^2)=(eta(2+eta))/((1+eta)^2)` or `v=csqrt((eta(2+eta))/((1+eta)^2))=(csqrt(eta(2+eta)))/(1+eta)`

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The density of a stationary body is equal to `rho_0`. Find the velocity (relative to the body) of the reference frame in which the density of the body is `eta=25%` greater than `rho_0`.

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