Schild’s Ladder

Appearance of the Border

Apparent shape of border

The border of the novo-vacuum is a sphere expanding at half the speed of light. Its apparent size in the sky is affected by the fact that when you look away from the closest point on the border, you’re looking back further in time, and seeing it when it was smaller.

In the diagram to the right, the black circles show the actual size of the border at five different times, while the blue curves show the apparent size and shape according to the stationary observer marked on the diagram, who has to wait for light from the border to arrive. (The mathematical form for these curves can be found by requiring that twice the distance from the centre of the border to a point on the curve, plus the distance from that point to the observer, is equal to the time elapsed since the creation of the novo-vacuum.)

The dashed grey lines show the apparent edge of the border; these are tangents to the blue curves. They show the path of starlight that grazed the border when it was smaller than its current size, allowing it to blot out much less of the sky than if it had been that size all along. Even at the latest time shown, when the border is right on top of the observer, it only occupies 120° of the view.

Doppler shift

The diagram to the left shows the increasing angular size of the border with time. The varied Doppler shift of the borderlight across the surface is indicated schematically; the exact blue shift factor ranges from √3 =1.732 at the centre down to 2/√3=1.1547 at the edge. The Doppler shift at the edge of the view remains the same as the border expands, because the light seen here is always emitted at exactly 90° from the direction of expansion (as measured in a reference frame moving along with the relevant portion of the border).

Valid HTML Valid CSS
Schild’s Ladder / Appearance of the Border / created Friday, 11 January 2002 / revised Saturday, 12 January 2002
If you link to this page, please use this URL:
Copyright © Greg Egan, 2002. All rights reserved.