Signature (- - +)

The lines radiating outward from (0,0,0) are geodesics, and the transverse "circles" are curves of constant geodesic parameter t .


Signature (+ - -)

The curves spiraling out and up from (0,0,0) are geodesics, and the transverse circles are curves of constant geodesic parameter t . From outside in, we have spacelike, null, and timelike geodesic surfaces, cut away to reveal their interiors. Thus less than one full period of each of these infinitely periodic surfaces is visible.


Signature (+ - -)

The middle part of one sheet in the upper half of the conjugate locus, in the same orientation (and coordinates) as the preceding picture, but with the infinite spike cut off. The black circle is null, above is timelike and below is spacelike. This fits in the hole in the center of the next picture.


Signature (+ - -)

The peripheral part of the same sheet in the upper half of the conjugate locus. All points are spacelike, and the height varies from 9.05 to 9.2 by 0.05; but superposition makes those numbers impossible to read.

The full conjugate locus consists of infinitely many sheets like these stacked up at regular intervals above the central plane, and their mirror images stacked down symmetrically below that plane. Each sheet extends to spacelike infinity asymptotically horizontally and the nested spikes all go up (down) to plus (minus) timelike infinity asymptotic to the timelike axis.


Between the two stacks of spiked sheets is the central part, which has the same timelike spikes up and down, but stays spacelike bounded, passing smoothly through the central plane.

Signature (+ - -)

The smooth, central part of the conjugate locus. The upper black circle is null, above is timelike (with the infinite spike cut off again), and below is spacelike as far as can be seen. In fact, the locus is symmetric through the central plane, and the lower null and timelike loci are out of the frame.


Signature (+ + -)

The lines radiating outward from (0,0,0) are geodesics, and the transverse "circles" are curves of constant geodesic parameter t . Shown are the null "cone," three timelike (spacelike for relativists) and two spacelike (timelike for relativists) surfaces.


Signature (+ + -)

The conjugate locus in the same orientation (and coordinates) as the preceding picture. Ragged edges are numerical artifacts.


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