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 $.