Consider a relativistically expanding shell . 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 will reach the observer at a time:
where is the distance between the shell's centre of expansion and the observer , is the shell's radius at a time and is the angle between the emission site and the line of sight to the observer. Plugging in , and shifting the arrival time by a constant amount: , we have:
It follows that: . Recalling that an ellipse is the shape defined by: , we see that the EATS is an ellipsoid with ellipticity: , a semi-major axis: and semi-minor axis: .