Consider a relativistically expanding shell  \Gamma \gg 1 . The equal arrival time surface (EATS) is the physical surface from which emitted photons arrive at the same time to the observer. A photon emitted at time  t will reach the observer at a time:

 T=t+\frac{D-R(t)cos \theta}{c}

where  D is the distance between the shell's centre of expansion and the observer ,  R(t) is the shell's radius at a time  t and  \theta is the angle between the emission site and the line of sight to the observer. Plugging in  R(t)=\beta c t , and shifting the arrival time by a constant amount:  T_0=T-\frac{D}{c} , we have:

 T_0=\frac{R}{\beta c}-\frac{R cos \theta}{c} .

It follows that:  \beta c T=R(1-\beta cos \theta) \rightarrow R=\frac{\beta c T}{1-\beta cos \theta}. Recalling that an ellipse is the shape defined by:  R=\frac{a(1-\epsilon^2)}{1-\epsilon cos \theta} , we see that the EATS is an ellipsoid with ellipticity:  \epsilon=\beta , a semi-major axis:  a=\gamma^2 \beta c T_0 and semi-minor axis:  b=\beta c T_0 .