The many weapons of general relativity

There are many objects used in either thought experiments or actual ones in general relativity, such as stars, spaceships, clocks, etc. For some reason, there seems to be a variety of weapons as well, mostly related to the weirder topics on the matter.

One of the only one that isn't a particularly odd topic is in John Synge's 1960 book, "Relativity : The general theory", which contains the so-called ballistic suicide problem :

The usual problem of ballistics is to aim a projectile so that it hits someone else. Here we consider the ballistic suicide problem : the projectile is to hit the projector himself!
However perverse such a problem may be sociologically, it is a neat problem in relativity, because there are only two observations and both are made by the same observer. Moreover it forces us to realize that although the trajectory of a projectile fired straight upward seems sharply curved at the top, from a spacetime standpoint it is as straight as possible (geodesic).

Ballistic problems in general relativity are fairly rare, but this one has the merit to exist.

Further on, there are many gun-related problems dealing with the notion of causality and closed timelike curves, faster-than-light propagation and advanced waves, all related to the famous grandfather paradox. While the grandfather paradox itself goes back in science fiction all the way back to the 1930's, its appearance in physics in a gun-related manner seems to be in Feynman and Wheeler's 1949 paper, "Classical electrodynamics in terms of direct interparticle action", which involved Feynman and Wheeler's emitter-absorber theory with advanced waves.

As advanced waves can produce causality violations, an example to illustrate the possible inconsistency problems is given :

To formulate the paradox acceptably, we have to eliminate human intervention. We therefore introduce a mechanism which saves charge $a$ from a blow at 6 p.m only if this particle performs the expected movement at 8 a.m. (Fig. 1). Our dilemma now is this: Is $a$ hit in the evening or is it not. If it is, then it suffered a premonitory displacement at 8 a.m. which cut off the blow, so $a$ is not struck at 6 p.m.! If it is not bumped at 6 p.m. there is no morning movement to cut off the blow and so in the evening $a$ is jolted!
Fig. 1. The paradox of advanced effects. Does the pellet strike $X$ at 6 p.m.? If so, the advanced field from $A$ sets $B$ in motion at 1 p.m., and $B$ moves $A$ at 8 a.m. Thereby the shutter $TS$ is set in motion and the path of the pellet is blocked, so it cannot strike $X$ at 6 p.m. If it does not strike $X$ at 6 p.m., then its path is not blocked at 5.59 p.m. via this chain of actions, and therefore the pellet ought to strike $X$

To resolve, we divide the problem into two parts: effect of past of $a$ upon its future, and of future upon past. The two corresponding curves in Fig. 2 do not cross. We have no solution, because the action of the shutter on the pellet, of the future on the past, has been assumed discontinuous in character.
Fig. 2. Analysis and resolution of the paradox of advanced eftects. The action of the shutter on the pellet—the interaction of past and future — is continuous (dashed line in diagram) and the curves of action and reaction cross. See text for physical description of solution.

While this paper does not deal directly with general relativity, it was an inspiration for further examples dealing with closed timelike curves, such as Clarke's paper "Time in general relativity" :

To see how this is so, consider the case already cited of a person who meets his former self in circumstances in which, if physics were normal, he would be able to shoot him. Then, as a preliminary step in the analysis, let us replace the complex human being by a simple automaton which nonetheless exhibits the abnormal physics referred to. This apparatus is to consist of a gun, a target, and a shutter so arranged that the impact of a bullet on the target will trigger the shutter so as to move in front of the gun. It pursues a causality-violating curve in spacetime in such a way that two points on the object's world line $A$ and $B$, with $B$ later in the object's history than $A$, are physically contemporaneous and disposed as in Figure 1 so that the gun at $B$ is aimed at the target at $A$ and the shutter is initially up at $A$.

Figure 1

Suppose now that the machine "shoots its former self" : the gun at $B$ is fired, either by an automatic timing mechanism or by the intervention of a human being making a conscious decision. If the shutter in $B$ were still up, the bullet would strike the target at $A$, which would cause the shutter in $B$ to be down, a contradiction. But if the shutter were down in $B$, then the bullet would be stopped, the target $A$ would not be hit, and the shutter in $B$ should still be up: the shutter is up if, and only if, it is down; the situation is logically impossible.

The though experiment is later found in Kriele's "Spacetime : Foundations of general relativity and differential geometry" :

At a first glance, the possibility of "free will" seems to be at the center of the issue. However, following (Wheeler and Feynman 1949) Clarke (1977) has re-formulated the thought experiment in terms of a simple machine and has argued that the thought experiment is fallacious : Assume that there is a gun directed at a target in spacetime. This target is connected with a shutter which, if closed, blocks off the path between the gun and the target : If the gun is triggered, the bullet will hit the target which in turn will cause the shutter to fall. A second shot will now be blocked by the shutter and therefore cannot hit the target (c.f. Fig. 8.1.3). Now assume that the configuration is located in a region with causality violation such that the shutter falls along a closed timelike curve so that it blocks the bullet before the gun has been triggered. Again we seem to arrive at a contradiction : If the shutter is open the bullet can hit the target. But the target closes the shutter which in turn blocks the path of the bullet.
A gedanken experiment to disprove causality violation

A slightly more radical version of this paradox was made in Novikov's paper "Time machine and self-consistent evolution in problems with self-interaction", in which a ball containing a bomb (exploding upon contact) is sent through a wormhole.

The initial data are arranged in such a way that the ball enters mouth $B$, emerges from mouth $A$ in the past, continues the motion and arrives at the point $Z$ just in time to collide with the "younger" version itself. This encounter leads to the explosion. We did not take into account the influence of the future on the past before the ball entered the mouth $B$, and this is the reason for the "paradox."

But there is a self-consistent evolution, as shown in Fig. 6. The initial data are the same as in Fig. 5, but before reaching the point Z it meets the fragment of the explosion of itself. This fragment hits the ball and it is the cause of the explosion, the fragments of the ball fly in all directions with velocities much larger than the velocity of the ball. Some of them fly into mouth Band emerge from mouth $A$ in the past. One can show that they will continue to fly in practically all directions from mouth A, because they have different impact parameters when they few into mouth $B$. One of the fragments from mouth $A$ crosses the trajectory of the ball at the point $Z'$ exactly at that moment when the ball arrives at the same point $Z'$. This fragment is the cause of the explosion of the ball. The consequence of the explosion (the fragment) is the cause of the explosion.

Fig. 5. The self-inconsistent evolution in the problem of a ball with a bomb.
Fig. 6. The self-consistent evolution in the same problem as Fig. 5.

A variation on this experiment follows as

Now let us consider the problem which is a more complicated version of the problem of the preceding section. The problem is the following (see Figs. 7—9). Let us suppose that there is the ball with a bomb and a radio transmitter (see Fig. 7), which gives a directed beam. The fuse explodes the bomb if, and only if, it is irradiated by the beam of such a radio transmitter from a distance of, say, $30\ m$ (see Fig. 7).

The self-inconsistent evolution is shown in Fig. 8. The "younger" ball explodes, on being irradiated by the radio transmitter of the "older" ball after it comes from the future.

Fig. 7. A billard ball with a bomb and a radio transmitter.

Now a fragment of the explosion cannot be the cause of the explosion and at first glance, the problem of constructing a self-consistent evolution looks insoluble, but that is not the case.

In Fig. 9 one can see the self-consistent evolution. Before reaching the mouth $B$ the "younger" ball encounters its "older" self from the future but with a change orientation of the radio transmitter (in fact the "younger" ball with the radio transmitter rotates after the point $Z$, and the "older" one rotates also). Now the fuse is not irradiated by the radio transmitter and there is no reason for the explosion. The inelastic collision of the "older" and the "younger" versions of the ball leads to a change in the orientation of the radio transmitters of both balls (rotation of the balls ) and drives both of them into slightly altered trajectories. Self-consistent evolution without an explosion is possible.

Fig. 8. The self-inconsistent evolution in the problem of a ball with a bomb and a radio transmitter.
Fig. 8. The self-consistent evolution in the same problem as in Fig. 8.

There are many such occurences of the grandfather paradox here and there in the literature, but those are some of the most famous instances. Some more cases can be found for instance in Earman's "Bangs, Crunches, Whimpers and Shrieks".

Beyond thought experiments, they also do pop up in proposals of actual experiments. Eric Davies, while investigating the Bertotti-Robinson spacetime in his paper "Interstellar Travel Through Magnetic Wormholes" (or as they reference it, the Levi-Civitta spacetime), which was suspected to be a wormhole-like solution by Claudio Maccone (this was later shown to be incorrect). As the Bertotti-Robinson spacetime corresponds to a homogeneous electric or magnetic field, the experimental proposal for the largest effect was to use a nuclear weapon to produce the largest possible magnetic field.

It will be necessary to consider advancing the state-of-art of magnetic induction technologies in order to reach static field strengths that are $> 10^9 - 10^{10}$ Tesla. Extremely sensitive measurements of $c$ at the one part in $10^6$ or $10^7$ level may be necessary for laboratory experiments involving field strengths of $\approx 10^9$ Tesla. Magnetic induction technologies based on nuclear explosives/implosives may need to be seriously considered in order to achieve large magnitude results. An order of magnitude calculation indicates that magnetic fields generated by nuclear pulsed energy methods could be magnified to (brief) static values of $\approx 10^9$ Tesla by factors of the nuclear-to-chemical binding energy ratio ($\approx 10^6$).
Posted on 2018-03-13 15:28:04
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