The Only Maximally Extended, Future-directed, Null and Timelike Geodesics in Gödel Spacetime are Confined to a Submanifold. Drunken Risibility.

Let γ1 be any maximally extended, future-directed, null geodesic confined to a submanifold N whose points all have some particular z ̃ value. Let q be any point in N whose r coordinate satisfies sinh2r = (√2 − 1)/2. Pick any point on γ1. By virtue of the homogeneity of Gödel spacetime, we can find a (temporal orientation preserving) global isometry that maps that point to q and maps N to itself. Let γ2 be the image of γ1 under that isometry. We know that at q the vector (t ̃a + kφa) is null if k = 2(1 + √2). So, by virtue of the isotropy of Gödel spacetime, we can find a global isometry that keeps q fixed, maps N to itself, and rotates γ2 onto a new null geodesic γ3 whose tangent vector at q is, at least, proportional to (t ̃a + 2(1 + √2)φa), with positive proportionality factor. If, finally, we reparametrize γ3 so that its tangent vector at q is equal to (t ̃a + 2(1 + √2)φa), then the resultant curve must be a special null geodesic helix through q since (up to a uniform parameter shift) there can be only one (maximally extended) geodesic through q that has that tangent vector there.

The corresponding argument for timelike geodesics is almost the same. Let γ1 this time be any maximally extended, future-directed, timelike geodesic confined to a submanifold N whose points all have some particular z ̃ value. Let v be the speed of that curve relative to t ̃a. (The value as determined at any point must be constant along the curve since it is a geodesic.). Further, let q be any point in N whose r coordinate satisfies √2(sinh2r)/(cosh2r) = v. (We can certainly find such a point since √2 (sinh2r)/(cosh2r) runs through all values between 0 and 1 as r ranges between 0 and rc/2) Now we can proceed in three stages, as before. We map γ1 to a curve that runs through q. Then we rotate that curve so that its tangent vector (at q) is aligned with (t ̃a + kφa) for the appropriate value of k, namely k = 2 √2/(1 − 2 sinh2r). Finally, we reparametrize the rotated curve so that it has that vector itself as its tangent vector at q. That final curve must be one of our special helical geodesics by the uniqueness theorem for geodesics.

The special timelike and null geodesics we started with – the special helices centered on the axis A – exhibit various features. Some are exhibited by all timelike and null geodesics (confined to a z ̃ = constant submanifold); some are not. It is important to keep track of the difference. What is at issue is whether the features can or cannot be captured in terms of gab, t ̃a, and z ̃a (or whether they make essential reference to the coordinates t ̃, r, φ themselves). So, for example, if a curve is parametrized by s, one might take its vertical “pitch” (relative to t ̃) at any point to be given by the value of dt ̃/ds there. Understood this way, the vertical pitch of the special helices centered on A is constant, but that of other timelike and null geodesics is not. For this reason, it is not correct to think of the latter, simply, as “translated” versions of the former. On the other hand, the following is true of all timelike and null geodesics (confined to a z ̃ = constant submanifold). If we project them (via t ̃a) onto a two-dimensional submanifold characterized by constant values for t ̃ as well as z ̃, the result is a circle.

Here is another way to make the point. Consider any timelike or null geodesic γ (confined to a z ̃ = constant submanifold). It certainly need not be centered on the axis A and need not have constant vertical pitch relative to t ̃. But we can always find a (new) axis A′ and a new set of cylindrical coordinates t ̃′, r′, φ′ adapted to A′ such that γ qualifies as a special helical geodesic relative to those coordinates. In particular, it will have constant vertical pitch relative to t ̃′.

Let us now consider all the timelike and null geodesics that pass through some point p (and are confined to a z ̃ = constant submanifold). It may as well be on the original axis A. We can better visualize the possibilities if we direct our attention to the circles that arise after projection (via t ̃a). The figure below shows a two-dimensional submanifold through p on which t ̃ and z ̃ are both constant. The dotted circle has radius rc. Once again, that is the “critical radius” at which the rotational Killing field φa is null. Call this dotted circle the “critical circle.” The circles that pass through p and have radius r = rc/2 are projections of null geodesics. Each shares exactly one point with the critical circle. In contrast, the circles of smaller radius that pass through p are the projections of timelike geodesics. The figure captures one of the claims – namely, that no timelike or null geodesic that passes through a point can “escape” to a radial distance from it greater than rc.

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Figure: Projections of timelike and null geodesics in Gödel spacetime. rc is the “critical radius” at which the rotational Killing field φa centered at p is null

Gödel spacetime exhibits a “boomerang effect.” Suppose an individual is at rest with respect to the cosmic source fluid in Gödel spacetime (and so his worldline coincides with some t ̃-line). If that individual shoots a gun at some point, in any direction orthogonal to z ̃a, then, no matter what the muzzle speed of the gun, the bullet will eventually come back and hit him (unless it hits something else first or disintegrates).

A Time Traveler in Gödel Spacetime

Given any two points p and q in Gödel spacetime, there is a smooth, future-directed timelike curve that runs from p and q. (Hence, since we can always combine timelike curves that run in the two directions and smooth out the joints, there is a smooth, closed timelike curve that contains p and q.)

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A time traveler in Gödel spacetime can start at any point p, return to that point, and stop off at any other desired point q along the way. To see why this holds, consider the figure above. It gives, at least, a rough, qualitative picture of Gödel spacetime with one dimension suppressed. We may as well take the central line to be the axis A and take p to be a point on A. (By homogeneity, there is no loss in generality in doing so.) Notice first that given any other point p′ on A, no matter how “far down,” there is a smooth, future-directed timelike curve that runs from p to p′. We can think of it as arising in three stages. (i) By moving “radially outward and upward” from p (i.e., along a future-directed timelike curve whose tangent vector field is of the form t ̃a + αra, with α positive), we can reach a point p1 with coordinate value r > rc. At that radius, we know, φa is timelike and future-directed. So we can find an ε > 0 such that (−εt ̃a + φa) is also timelike and future-directed there. (ii) Now consider the maximally extended, future-directed timelike curve γ through p1 whose tangent is everywhere equal to (−εt ̃a + φa) (for that value of ε). It is a spiral-shaped curve of fixed radius, with “downward pitch.” By following γ far enough, we can reach a point p2 that is well “below” p′. Now, finally, (iii) we can reach p′ by working our way upward and inward from p2 via a curve whose tangent vector is the form t ̃a + αra, but now with α negative. It remains only to smooth out the “joints” at intermediate points p1 and p2 to arrive at a smooth timelike curve that, as required, runs from p to p′.

Now consider any point q. It might not be possible to reach q from p in the same simple way we went from p to p1 – i.e., along a future-directed timelike curve that moves radially outward and upward – p might be too “high” for that. But we can get around this problem by first moving to an intermediate point p′ on A sufficiently “far down” – we have established that that is possible – and then going from there to q.

Other interesting features of Gödel spacetime are closely related to the existence of closed timelike curves. So, for example, a slice (in any relativistic spacetime) is a spacelike hypersurface that, as a subset of the background manifold, is closed. We can think of it as a candidate for a “global simultaneity slice.” It turns out that there are no slices in Gödel spacetime. More generally, given any relativistic spacetime, if it is temporally orientable and simply connected and has smooth closed timelike curves through every point, then it does not admit any slices.