# UFOs and skepticism

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On the subject of FTL and causality violation, consider a simple thought experiment, assuming that FTL exists.

There are two participants, A and B, both of whom have FTL transmitters. Participant A sends an FTL signal to B, recording the time of transmission as shown on a clock in his own reference frame. By a pre-arranged plan, as soon as B receives the signal he sends an FTL signal back to A to confirm receipt of A's signal. A records the time at which he receives B's signal, as shown on A's clock. Does anyone dispute that the time recorded by A for his receipt of B's return signal will be later than the time he records for his original signal to B? If so, then there is no violation of causality in A's reference frame. If there is an apparent violation of causality in some other reference frame, I would argue that either the reference frame or the method of calculating time is inappropriate for recording FTL events. It would be like using a method based on the speed of sound, assumed to be the highest possible speed, for recording signals transmitted by light.

I am not for a moment denying the vast amount of evidence that the speed of light is constant, and is the highest physically possible speed, in all inertial reference frames, and that FTL signals or travel are therefore physically impossible. I am just questioning that argument that FTL is impossible because it would violate causality. I think that argument only works in the conceptual framework of Special Relativity, which assumes as a postulate that the speed of light is a constant and limiting velocity. If that assumption is empirically false, all bets are off.

It would be like using a method based on the speed of sound, assumed to be the highest possible speed, for recording signals transmitted by light.
Or using the famous light cone diagrams and happily extend your drawing outside the light cone of an inertial frame, which is outside the area where SR is applicable (the results of the Lorenz transformation become imaginary there, which makes no sense).

Picture does not seem to load, here is the link: https://www1.phys.vt.edu/~takeuchi/relativity/notes/ArrowApple.gif

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Our current models in physics have the ability to predict, yes. But not everything they predict has been experimentally verified. New experiments may show errors in their predictions, which will lead to new models in the future. So only our current set of experimental observations is "scientifically sacred", not our current models and everything they predict regardless of experimental verification.

Agreed on all counts. Doesn't really contradict with what I wrote earlier.

Or using the famous light cone diagrams and happily extend your drawing outside the light cone of an inertial frame, which is outside the area where SR is applicable (the results of the Lorenz transformation become imaginary there, which makes no sense).

Not so, SR applies just as much to space as it does to time, and the Lorenz transformation maps space-like vectors onto space-like vectors exactly how it maps time-like vectors onto time-like vectors (and light-like vectors remain unchanged, of course).

Not so, SR applies just as much to space as it does to time, and the Lorenz transformation maps space-like vectors onto space-like vectors exactly how it maps time-like vectors onto time-like vectors (and light-like vectors remain unchanged, of course).
Any worldline you draw outside of the light cone means v>c. Now let's look at what comes out of the Lorenz tranformation in that case:

(Source: https://en.m.wikipedia.org/wiki/Lorentz_transformation)

Since the Lorenz factor becomes an imaginary number (a multiple of i, with i-square equals -1), both time and distance become an imaginary number, which does not make sense at all.
That is why Wikipedia states: "The value of v must be smaller than c for the transformation to make sense."

So any worldline you draw outside of the light cone lies outside of the applicable area of SR, and in the "twilight zone". You cannot casually use them to demonstrate a violation of causality, which is what many scientists tend to do (!).

Any worldline you draw outside of the light cone means v>c. Now let's look at what comes out of the Lorenz tranformation in that case:

View attachment 57263
(Source: https://en.m.wikipedia.org/wiki/Lorentz_transformation)

You cannot correct my usage of the term worldline, nor of my usage of v, because I do not mention worldlines, nor do I mention any v. The transformation acts on the whole plane, which includes both the time-like and the space-like vectors.

The transformation acts on the whole plane, which includes both the time-like and the space-like vectors.
The light cone diagrams are just a graphical method of solving the Lorentz equations. If the equations do not make sense for v>c, then the corresponding graphical method (applied in the picture at post #42 above) does not either.

Any worldline you draw outside of the light cone means v>c. Now let's look at what comes out of the Lorenz tranformation in that case:

View attachment 57263
(Source: https://en.m.wikipedia.org/wiki/Lorentz_transformation)

Since the Lorenz factor becomes an imaginary number (a multiple of i, with i-square equals -1), both time and distance become an imaginary number, which does not make sense at all.
That is why Wikipedia states: "The value of v must be smaller than c for the transformation to make sense."

So any worldline you draw outside of the light cone lies outside of the applicable area of SR, and in the "twilight zone". You cannot casually use them to demonstrate a violation of causality, which is what many scientists tend to do (!).
Of course, Lorentz transformations make no sense when the velocity is greater than c. That is understood. That is why the argument that gives violations of causality from superluminal motion makes no use of such a thing!

The argument goes like this: imagine you have some 'object' moving with some speed v0 > c. In these vague enough terms, this is not yet an impossibility: the 'object' might not be a physical object, but rather a shadow, the end of a laser pointer beam, etc. Say we're in a reference frame where the Earth is at rest at the origin (which I'll call "the Earth's reference frame"), and that at t=0 the object passes x=0. Then at t=4s, it passes by Alpha Centauri, x=~4 light years.

Clearly, I cannot describe what happens from the perspective of the object. As far as I know, such a perspective might not even exist or make sense as a concept. But, can I describe what happens from the perspective of other, subluminal reference frames? You bet. Consider what happens from the perspective of Alice, who's on a spaceship heading towards Alpha Centauri with a speed of v=0.98c (γ ≈ 5), passing by Earth (x'=x=0) at t'=t=0 (following wikipedia's convention, primed coordinates are in Alice's reference frame). When does the object reach Alpha Centauri, according to her? We only need one of the Lorentz transformation equations, but I'll write them both for completeness:

t' = γ ((4 s) - v (4 ly) / c²)
x' = γ ((4 ly) - v (4 s))

t' ≈ 5 * ((4 s) - 0.98 * (4 ly) / c) ≈ 5 * (4 s - 0.98 * 4 years) ≈ -19.6 years
x' ≈ 5 * ((4 ly) - 0.98c * (4 s)) ≈ 5 * (4 ly - 3.92 ls) ≈ 20 ly

So Alice would see the object arrive at Alpha Centauri almost 20 years before it departed Earth.

This is not that surprising, in fact, it's just the familiar argument about relativity of simultaneity wearing a fake mustache. The Lorentz transformations, as FatPhil said, map spacelike ("faster than light") intervals to spacelike intervals and timelike ("slower than light") intervals to timelike intervals, but only in the latter case is the relative order of events preserved. Therefore, the only kinds of trajectories allowed to real particles and objects carrying information are timelike (and lightlike).

I emphasize that in this argument I only made use of special relativity in regimes where it is applicable, well understood, and tested -- no superluminal Lorentz transformations here. If we take the humble interpretation that the "object" is the beam of a laser pointer, all this holds and is perfectly meaningful, yet causes no paradox: just because someone points a laser pointer at me doesn't mean I can use it to send a message to the next victim.

But say that it is a real object, carrying information. You might posit, as DavidB66 did, that Alice is just "wrong" here -- that the object really arrives after it departed, and that any perception to the contrary is merely some kind of perspective illusion. Well, as long as you're unwilling to reject special relativity altogether, that doesn't help either. Consider the following scenario:

1. Bob, on Earth, sends a superluminal (1 light-year / second) message to Alpha Centauri.
2. Alice, now at Alpha Centauri (x = 4 ly), moving with the same velocity of 0.98c along the direction from Earth to Alpha Centauri, receives the message at t' ≈ -19.6 years.
3. Alice replies immediately by superluminal message (1 light-year / second from her perspective), which arrives at Earth at t' ≈ -19.6 years. When's that, according to Bob? We use the inverse transformation:

t = γ (t' + v x' / c²)
x = γ (x' + v t')

Earth is at x = 0:
0 = γ (x' + v t') -> x' = -vt' (as expected)

Plugging into the equation for t,
t = γ (t' - v² t' / c²) = γ (1 - v² / c²) t' = t' / γ = -19.6 years / 5 = -3.92 years

Bob receives the message 3.92 years before he sent it.

This is now a bona-fide closed timelike curve, a fact which is independent of reference frame and agreed upon by all observers (though Alice and Bob disagree on which is leg is the one that's backwards in time).

Note also that it doesn't matter what the putative mechanism might be for the superluminal messaging -- it doesn't matter if it's a warp drive, a wormhole, tachyons, some kind of ansible, whatever. As long as spacetime far away from the object remains undeformed, special relativity applies and a version of this argument can be constructed. The key thing is the ability to do superluminal round trips while preserving the principle of relativity, that is, the idea that there's nothing special about Earth's reference frame when compared to Alice's.

The last sentence also indicates the limits of applicability for this argument. If the principle of relativity fails to hold, it may be impossible to traverse one or both of those superluminal legs, which could save causality after all. For example, if there's some secret reference frame with respect to which superluminal travel takes place, we're morally back to Newton: time is once more absolute. Or maybe you forbid superluminal travel unless it's going from left to right, etc. The important point here is that in all cases we're rejecting relativity in a pretty big way. It's not just some illusion or cosmic clerical error.

Hence the dictum: relativity, causality, FTL. Pick at most two.

I'd like this explained, please.
1. Bob, on Earth, sends a superluminal (1 light-year / second) message to Alpha Centauri.
2. Alice, now at Alpha Centauri (x = 4 ly), moving with the same velocity of 0.98c along the direction from Earth to Alpha Centauri, receives the message at t' ≈ -19.6 years.
3. Alice replies immediately by superluminal message (1 light-year / second from her perspective), which arrives at Earth at t' ≈ -19.6 years. When's that, according to Bob?
The naive answer is, 8 seconds later than he sent his message.

An Alpha Centaurian will see the superluminal message at t"=0, send one back in the direction from whence it came that looks like empty space now, and over the subsequent 4 years AC's astronomers track both messages and Earth towards the rendezvous point, where Bob's message is seen to arrive 8 seconds before Alice's message. It takes a very exact knowledge of the distance to Earth (7 significant digits) at rendezvous time to even determine the direction of travel for both messages.

Alice, on the other hand, is travelling away from Earth at 0.98c (starting at AC), so from AC's perspective she has to wait 200 years to see that rendezvous, and Alice should see the messages "arrive" at Earth 400s (AC seconds) apart, in the correct, causality-preserving order.

Why do the Lorenz transformations affect causality from Alice's perspective?
Does it matter whether Alice sends the return message, or someone on AC does?

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Hence the dictum: relativity, causality, FTL. Pick at most two.
I think we need a bigger dictum: relativity, causality, FTL, facts. Pick at most three.

If facts - a.k.a. empirical evidence - showed beyond doubt that FTL communication was possible, then something else would have to give. Empirical evidence trumps all other considerations. There might be a choice between giving up relativity and giving up causality, in which case (as I said before) I would prefer to give up relativity. But would it need to be as drastic as that? Newtonian mechanics is strictly incompatible with relativity, but Newtonian mechanics is still an excellent approximation in many cases. Likewise STR is strictly incompatible with GTR, but an excellent approximation when gravitational fields are weak (and, if I understand correctly, the approximate validity of STR 'at the limit' is assumed in deriving the wider theory of GTR itself). If FTL proved to be possible, this might involve some hitherto unknown extension of physics, such as the reality of more than three spatial dimensions, in which case relativity might still hold good in the boring old case of three (+ time).

The most important word in the last sentence was 'if'. At present we have no evidence for FTL travel or communication, other than the very dubious reports of UFOs, and a great deal of evidence that light-speed is a fundamental physical limit.

I am thinking if extra terrestrials came by for a visit they most like would be interested in studying the planet... geology, life forms, weather and so forth. As such I doubt a single "fly by" would satisfy that mission. So... I expect if there would be multiple sightings in numerous places in a reasonably compressed time frame. I suppose they could also park some place and do their surveillance from there. And for sure things would be very different as little as a few hundred years ago. And extremely different 3 or 4 thousand years ago. Weather survey might not vary much but the development of human civilization really is time sensitive.

This is now a bona-fide closed timelike curve, a fact which is independent of reference frame and agreed upon by all observers (though Alice and Bob disagree on which is leg is the one that's backwards in time).
The answer is simple: I simply move the origin of Bob's coordinate system to Alpha Centauri which puts Bob at coordinate -x and Alpha Centauri at x=0. This yields t'= γ t

Of course this does not work. Why can't I arbitrarily choose the origins of these inertial systems? Because the times in the Lorentz transformation are the times at which an observer in the origin of the corresponding inertial frame actually observes an event. This can be easily understood if you look at the tranformation for time: t' = γ (t - v x / c²). If I could freely move Bob's inertial system's origin up and down without changing Bob's position, I could freely choose Alice's observation time of an event without changing anything to the situation!

The observer moving in the positive x direction will always observe an event in front of him earlier than the observer at (relative) rest (save for the γ value, representing time dilation), and the further away the event takes place (the bigger x is), the bigger the time difference will be. This is because the moving observer has more time to move closer to the event while the light carrying information about the event is moving towards him if the event takes place at a larger distance.

This means that the value for t you have to fill in is the time at which Bob observes the event (which is 4 years plus 4 seconds, NOT 4 seconds). This means Alice, in turn, observes it after 0.1 Alice-years, NOT -19 years... This makes a lot more sense, since it will take Alice just over 4 Bob-years to reach Alpha Centauri.
In the inverse transformation, you'll have to fill in the time when Alice actually observes her return signal arriving at Bob for t', NOT 1. You'll then see that the causality violation is no longer there.

Edit: removed some inaccuracies

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I think we need a bigger dictum: relativity, causality, FTL, facts. Pick at most three.

If facts - a.k.a. empirical evidence - showed beyond doubt that FTL communication was possible, then something else would have to give. Empirical evidence trumps all other considerations.

The part in bold is the salient point for this discussion. As I've reiterated, any and all scientifically viable alternative theories that either tweak or fundamentally change received physical theories which successfully predict a substantial increasingly growing set of evidence (empiria / observations / measurement outcomes), must be at least as successful in predicting the same ever-growing set of evidence by the virtue of sound logical/mathematical deduction. This basic scientific requirement of new revolutionary theories in no wise contradicts the fact that even the most empirically robust amongst the received theories remain incompatible with a number of anomalous observations and measurement outcomes and hence are, obviously, not theories of everything, or completed works of perfection under museum glass never to be broken. This strawman of the skeptics' rigid ultra-conservative defensiveness for received models must therefore be repeatedly called out.

There might be a choice between giving up relativity and giving up causality, in which case (as I said before) I would prefer to give up relativity.

I tend to agree. However, either way, giving up causality in the specific sense intended by relativity is not giving up causality entirely as a physical let alone philosophical principle. In relativity causes and effects are events within spacetime under the additional constraint of the speed of light.

Philosophically (including the philosophy of physics) the question 'what causes spacetime?' and 'what causes light/electromagnetism?' (irrespective of whether time or electromagnetism has an infinite past or not) is a perfectly reasonable one and hence the notion that causality is necessarily a relation in time is both incorrect and restrictive. For instance, Newton's law of gravitation proposed by his NGT was not a physical cause in the sense of an event dependent on another event preceding it along a chronological causal chain. The law of gravitation (albeit disproven as a 'force' by Einstein) was an idea of a universal law-like force causing attraction between particles which does not operate/occur earlier in time from the actual motion of the particles, or locally. It was simultaneous and universal. Many surviving laws of physics could be understood similarly in a causal sense which is completely time-independent. Similarly, the sun can be reasonably regarded as the cause of solar radiation whilst it's impossible to tell if or when the sun exactly precedes solar radiation in time. And yet the sun is obviously an entity that precedes and causes an electromagnetic effect.

Leibniz's powerful (albeit still debated) Principle of Sufficient Reason (PoSR) was philosophically deliberately non-specific as to the types of reasons or causes implied in the maxim that 'everything must be preceded by a cause'. In other words, any entity, property or behaviour can theoretically have both deterministic or indeterministic causes ('law-likeness' vs. 'chanciness'). Furthermore, the entity, or its property or behaviour, may have causes both/either within the entity (self-caused) or outside the entity (other-caused). And yet furthermore, causes may be total causes (causes for the very existence of an entity) or partial causes (causes for a particular property or behaviour of an entity). And yet furthermore, and closely related, causes may be composite (many causes) or simple (single cause). Even the foregoing list is not exhaustive in terms of all logically possible types of causes.

However, all of the above types of causes are worth exploring in any exercise purporting to describe itself as 'scientific'. What PoSR squarely disagrees with is the philosophical alternative that an entity, property or a behaviour can be/occur without any cause whatsoever. This fundamental rule posited by the PoSR is often justified by appeal to the philosophical dictum ex nihilo nihil fit ("nothing comes out of nothing").

According to the dictum, while it's logically possible that there be an entity or a property that has no cause whatsoever, it's an absurd and unhelpful possibility. Hence, for instance, the known universe may be, under the PoSR, rationally explored either as having a sufficient reason within itself for its own existence (self-caused) or without itself (other-caused). But the third option that there is no sufficient reason for the universe to exist is regarded as absurd, for the question can always be reasonably asked: If the universe does not have a sufficient reason to exist at all, how come it exists nonetheless? Similarly, nothingness has no capacity to exist, and so nothing can come out of nothing. These intuitively powerful statements contribute to the continuing philosophical plausibility of the PoSR. The proponents of the third option (there is no sufficient reason/cause for existence) forbid the question 'why' or 'how' and insist on acceptance of existence as a brute fact.

But I realize I'm now digressing. Apologies. I'm sure you got what I was after.

The whole issue of FTL travel is based on human frailty. 300 years for a non-FTL trip to Alpha Centauri may seem to be an insurmountable barrier to us, but to an intelligent creature with a typical lifespan of 1000 years it would seem far less daunting. How difficult something appears to be is based on what you know your own capabilities to be. FTL is a fun topic to speculate on, but to a creature with a 100,000 year lifespan it might appear quite irrelevant to the question of interstellar travel.

The naive answer is, 8 seconds later than he sent his message.
There is a tacit assumption there that reference frames in which the Earth is at rest are special, and that a superluminal message will have the prescribed speed of 1 ly/yr in that reference frame. But a key postulate of relativity is that all reference frames are equivalent, and, unless we wish to reject it, we must consider the possibility that a superluminal message may have that prescribed speed in a different reference frame.

An Alcubierre warp drive is an example of a scheme in which we can set this up: since the backdrop is general relativity, we can set up Alcubierre warp bubbles in whatever reference frame we choose. But, a warp bubble moving at 1 ly / yr according to Alice will be seen as traveling backwards in time according to Bob.
Does it matter whether Alice sends the return message, or someone on AC does?
The answer, then, is yes: if Charlie, at the research station in orbit around Alpha Centauri, intercepts Bob's message and immediately replies, the reply will be received back on Earth at t = 8s. What causes the paradox is the fact that what Alice sees as a "harmless" superluminal signal is a backwards in time signal according to Charlie and Bob, and her perspective is perfectly valid.

What the round-trip is doing here is something analogous to what it does in the twins paradox, where bringing the twin home forces us to confront the consequences of special relativity by showing what happens according to the clocks on Earth, eliminating any possibility that time dilation is some weird coordinate illusion.

I have to say I couldn't understand what you were getting at with your middle two paragraphs though.

I think we need a bigger dictum: relativity, causality, FTL, facts. Pick at most three.

If facts - a.k.a. empirical evidence - showed beyond doubt that FTL communication was possible, then something else would have to give.
Then you'd pick FTL and relativity or FTL and causality -- or maybe just FTL, and throw out both of the others. What the dictum summarizes is something about what sets of ideas you can consistently hold simultaneously; it does not say anything about the reasons why you might hold said ideas.
Empirical evidence trumps all other considerations. There might be a choice between giving up relativity and giving up causality, in which case (as I said before) I would prefer to give up relativity. But would it need to be as drastic as that? Newtonian mechanics is strictly incompatible with relativity, but Newtonian mechanics is still an excellent approximation in many cases. Likewise STR is strictly incompatible with GTR, but an excellent approximation when gravitational fields are weak (and, if I understand correctly, the approximate validity of STR 'at the limit' is assumed in deriving the wider theory of GTR itself).
In the same way that a sphere looks like a plane if you are close enough to its surface, general relativity looks like special relativity if you look at small regions of curved spacetime. Special relativity is, in this sense, continuously deformable into general relativity, and Newtonian mechanics is continuously deformable into relativistic mechanics. This property is what justifies the former as a low-energy, approximate description of the latter. In contrast, should FTL be possible, we'd have to regard relativity as a sort of mirage, something that was never true in any sense. It's logically possible, but different from other examples of paradigm shifts we have encountered.
If FTL proved to be possible, this might involve some hitherto unknown extension of physics, such as the reality of more than three spatial dimensions, in which case relativity might still hold good in the boring old case of three (+ time).
For the purposes of this argument, all that matters are the laws in this set of dimensions we see. If, for instance, spacetime was warped a la Randall-Sundrum and you used that to shorten travel times by going through a "shortcut" in the warped dimensions, the argument above would apply.

The answer is simple: I simply move the origin of Bob's coordinate system to Alpha Centauri which puts Bob at coordinate -x and Alpha Centauri at x=0. This yields t'= γ t
Gotta be a little careful there -- if moving the origin changes the conclusion of an argument, what you have is not a "paradox" but a mathematical inconsistency and you have to throw the whole thing out. Seems doubtful that special relativity is inconsistent in this way, so it's worth checking if something was missed in the transformation. Let's say the origin x = 0 is at Alpha Centauri, so Earth is at x = -4 ly:

x = γ (x' + vt')
-4 ly = γ (x' + vt')
x' = -4 ly / γ - vt'
x'= -(vt' + 4 ly / γ).

Plugging into the equation for time,

t = γ (t' + v x' / c²)
t = γ (t' - v (vt' + 4 ly / γ) / c²)
t = γ (t' - (v/c) ((v/c) t' + 4 yr / γ))
t = γ t' (1 - (v/c)²) - γ (v/c) (4 yr / γ)
t = t' / γ - (v/c) (4 yr)

Up to an additive constant (a shift in time to correspond to the shift in space), this is the same answer from before. Relativity is still ok.
Of course this does not work. Why can't I arbitrarily choose the origins of these inertial systems? Because the times in the Lorentz transformation are the times at which an observer in the origin of the corresponding inertial frame actually observes an event.
That is incorrect. The Lorentz transformations map x and t coordinate values between different reference frames, analogously to how a rotation in the 2d plane maps x and y values between different coordinate systems. For example, a 90 degree counterclockwise rotation of my coordinate axes takes the coordinate point (x = 0, y = 1) to (x' = 1, y' = 0). It's a purely geometric transformation, a fact which is most evident when they are written in terms of hyperbolic functions,
ct' = γ (ct - (v/c) x)
x' = γ (x - (v/c) ct)

Define ϕ by sinh ϕ = γ (v/c); the relation cosh² ϕ - sinh² ϕ = 1 then gives
cosh ϕ = sqrt(1 + γ²(v/c)²)
cosh ϕ = sqrt(1 + (v/c)² / (1 - (v/c)²))
cosh ϕ = sqrt((1 - (v/c)²) / (1 - (v/c)²) + (v/c)² / (1 - (v/c)²))
cosh ϕ = γ.

Then
ct' = cosh ϕ ct - sinh ϕ x
x' = -sinh ϕ ct + cosh ϕ x

which looks very similar to a rotation matrix,
x' = cos θ x - sin θ y
y' = sin θ x + cos θ y

The Lorentz transformation is, quite literally, "just" a hyperbolic analogue of a rotation.

(incidentally, the hyperbolic angle ϕ is known as the "rapidity").

The point of this is to understand that Lorentz transformations are nothing more, nothing less than a mapping between different coordinate systems, and that nothing like a "light speed delay" is considered. You have to do it yourself if you want to take it into account. As they stand, the transformations tell you when and where events actually happen in the respective coordinate systems, not when this or that observer gets to detect them.

(For a fun consequence of the difference between coordinate transformations and detection times, see Penrose-Terrell rotations).

All that said, the light speed delay doesn't change much here. Take the round-trip example: there is no light speed delay at all, because Bob receives the superluminal message on Earth, at x = 0.

The point of this is to understand that Lorentz transformations are nothing more, nothing less than a mapping between different coordinate systems, and that nothing like a "light speed delay" is considered.
OK, let's put this trough the test then.
We leave Alice's coordinate system unchanged.
We leave the physical situation unchanged.
We only change the origin of Bob's coordinate system: In case 1 the origin lies with Bob, in case 2 it lies at Alpha Centauri.

If you are right, the same physical event would have to map onto the same coordinates in Alice's coordinate system for both cases.

For both cases:
- We measure time in years (y) and distance in light-years (ly), so c=1
- Alice passes Bob at v = 0.98c and maintains this speed towards Alpha Centauri
- At the moment when Alice passes Bob they reset their clocks to t=t'=0
- At t=0, Bob sends a superluminal signal with speed 1 light-year/second to Alpha Centauri
- Gamma = 1/sqrt(1-0,98^2) = 5.025
- The origin of Alice's coordinate system lies at Alice's position

We are interested in the (x, t) coordinates of event E, which is when Bob's superluminal signal hits Alpha Centauri, in both coordinate systems.

Case 1: The origin of Bob's coordinate system lies at Bob's position.
(x, t) of event E = (4 ly, 4 s) = (4 ly, 1.27E-7 y)
Now we apply the Lorentz formulas to find (x', t'), the coordinates of the same event in Alice's coordinate system:
x' = 5.025(4 - 0.98*1.27E-7) = 20.1 ly
t' = 5.025(1.27E-7 - 0.98*4) = -19,7 y

Case 2: The origin of Bob's coordinate system lies at Alpha Centauri (Bob's x-position is -4 ly).
(x, t) of event E = (0 ly, 4 s) = (0 ly, 1.27E-7 y)
Now we apply the Lorentz formulas to find (x', t'), the coordinates of the same event in Alice's coordinate system:
x' = 5.025(0 - 0.98*1.27E-7) = -6,25E-7 ly
t' = 5.025(1.27E-7 - 0.98*0) = 6,38E-7 y

So for case 1 we have (x', t') = (20.1 ly, -19.7 y)
And for case 2 we have (x', t') = (-6,25E-7 ly, 6,38E-7 y)

If you are right, and this is just a mapping between coordinate systems, both cases should yield exactly the same answers because:
- We did not change anything to Alice's speed, clock, or Alice's coordinate system.
- We did not change anything to the physical position and timing of event E from Alice's viewpoint.
Yet the answers differ...

if moving the origin changes the conclusion of an argument, what you have is not a "paradox" but a mathematical inconsistency and you have to throw the whole thing out.
No, the origin of inertial systems plays a key role here, since Special Relativity is a theory that predicts observations.
In terms of SR, case 1 is impossible. Bob cannot observe an event 4 ly away after 4 seconds. It should take at least 4 years. That is why the math falls apart here and yields these strange values for Alice.

- At the moment when Alice passes Bob they reset their clocks to t=t'=0
That means that for case 1: x=0,t=0 and x'=0, t'=0
And for case 2: x=-4,t=0 and x'=0,t'=0
The synchronization conditions are different.
Case 1: The origin of Bob's coordinate system lies at Bob's position.
(x, t) of event E = (4 ly, 4 s) = (4 ly, 1.27E-7 y)
Now we apply the Lorentz formulas to find (x', t'), the coordinates of the same event in Alice's coordinate system:
x' = 5.025(4 - 0.98*1.27E-7) = 20.1 ly
t' = 5.025(1.27E-7 - 0.98*4) = -19,7 y

Case 2: The origin of Bob's coordinate system lies at Alpha Centauri (Bob's x-position is -4 ly).
(x, t) of event E = (0 ly, 4 s) = (0 ly, 1.27E-7 y)
Now we apply the Lorentz formulas to find (x', t'), the coordinates of the same event in Alice's coordinate system:
x' = 5.025(0 - 0.98*1.27E-7) = -6,25E-7 ly
t' = 5.025(1.27E-7 - 0.98*0) = 6,38E-7 y

So for case 1 we have (x', t') = (20.1 ly, -19.7 y)
And for case 2 we have (x', t') = (-6,25E-7 ly, 6,38E-7 y)
I believe in case 2 you are also making the synchronization using the same condition as in case 1 (x=0,t=0) (x'=0,t'=0), which means "At the moment when Alice passes Alpha Centauri they reset their clocks to t=t'=0"

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OK, let's put this trough the test then.
We leave Alice's coordinate system unchanged.
We leave the physical situation unchanged.
We only change the origin of Bob's coordinate system: In case 1 the origin lies with Bob, in case 2 it lies at Alpha Centauri.

On Lorentz transformations (bold added):

The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.

On Lorentz transformations (bold added):

The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.
Yeah, I re-read the derivation of the Lorentz transformation by Einstein and somewhere along the line he uses v=x/t, where v is the velocity of the 'other' inertial system (the one moving relative to the one considered). This implies that their clocks must be reset to 0 at the moment when the origins of their inertial systems pass each other, and time is calculated from there on. So this basically fixes the origin of each inertial system.

Re-considering the Bob & Alice case, I think the causality paradox is caused by the fact that the hypothetical Superluminal Communication Device (SCD) switches inertial frames: The first signal is sent from Bob's inertial frame and the return signal from Alice's inertial frame. This causes a device sending superluminal signals (v>c) to be subjected to Lorentz transformations, which leads to strange results because these transformations were not intended for v>c. Yet the receiving SCD is forced to take over Alice's clock, which leads to the causality paradox.

A cleaner and more honest way to treat this case is to consider the SCD returning the message to be in Bob's inertial frame. Any Lorentz transformation involving the SCD is completely avoided then. In terms of the thought experiment this means Alice presses a button (or sends a short bluetooth signal) to a SCD installed by Bob an Alpha Centauri, and this SCD sends the return message. This return message arrives back at Bob after 4 'Bob-seconds'. The causality paradox disappears entirely now:

We do see that the return message arrives after 20,5 years on Alice's clock, so Alice will have to wait a while...

The whole issue of FTL travel is based on human frailty. 300 years for a non-FTL trip to Alpha Centauri may seem to be an insurmountable barrier to us, but to an intelligent creature with a typical lifespan of 1000 years it would seem far less daunting. How difficult something appears to be is based on what you know your own capabilities to be. FTL is a fun topic to speculate on, but to a creature with a 100,000 year lifespan it might appear quite irrelevant to the question of interstellar travel.

While there's no way to know, it seems possible to me that a 100,000-year lifespan creature would not be able to evolve sufficiently (not enough generations) to invent much technology at all, let alone space travel.

If a super-intelligent species extended its lifespan that long, similar problems arise but on the backend.

Yeah, I re-read the derivation of the Lorentz transformation by Einstein and somewhere along the line he uses v=x/t, where v is the velocity of the 'other' inertial system (the one moving relative to the one considered). This implies that their clocks must be reset to 0 at the moment when the origins of their inertial systems pass each other, and time is calculated from there on. So this basically fixes the origin of each inertial system.
Indeed, the Lorentz transformations are linear so they always map origin to origin, by definition. In fact, the synchronization condition you imposed is inconsistent with this property:

t' = γ (t - (v/c) (x / c)) = -γ (v/c) (-4 ly / c) ≈ 19.6 years
Re-considering the Bob & Alice case, I think the causality paradox is caused by the fact that the hypothetical Superluminal Communication Device (SCD) switches inertial frames: The first signal is sent from Bob's inertial frame and the return signal from Alice's inertial frame. This causes a device sending superluminal signals (v>c) to be subjected to Lorentz transformations, which leads to strange results because these transformations were not intended for v>c. Yet the receiving SCD is forced to take over Alice's clock, which leads to the causality paradox.
Quite right once more, the "switch in reference frames" is the cause of this strange conclusion, just like in the Twins' paradox.
A cleaner and more honest way to treat this case is to consider the SCD returning the message to be in Bob's inertial frame.
But you can't really fix it that easily. Let's say that our ansible has a restriction: you can only receive messages if you are at rest with respect to the signal source. Consider the following scheme:

1. Alice sends a superluminal message to Charlie, who's at rest near Alpha Centauri.
2. Charlie sends an ordinary radio message to Bob, at the exact instant when Bob is zooming by at 0.98c.
3. Charlie sends a superluminal reply toward Earth.
4. David, who's near Earth on a rocket also traveling 0.98c along the line from Earth to Alpha Centauri, receives the message.
5. David relays the message to Bob via ordinary radio.

The exact same calculations apply, despite the extra steps, and the causality paradox persists.

This is because, as I've been emphasizing, special relativity is not a theory of "observations" but rather one describing the geometry of spacetime itself. As such, it doesn't care how the tasks are apportioned between different actors, etc -- what matters is what path the information takes through spacetime. If there's no information (like if the path was traced by a laser pointer), the geometry is exactly identical, including backwards in time steps. But allowing actual information-carrying objects to traverse this path lets me send myself the lotto numbers or something, and that's ultimately where a "strange result" turns into a "causality violation".

Indeed, the Lorentz transformations are linear so they always map origin to origin, by definition. In fact, the synchronization condition you imposed is inconsistent with this property:

t' = γ (t - (v/c) (x / c)) = -γ (v/c) (-4 ly / c) ≈ 19.6 years

Quite right once more, the "switch in reference frames" is the cause of this strange conclusion, just like in the Twins' paradox.

But you can't really fix it that easily. Let's say that our ansible has a restriction: you can only receive messages if you are at rest with respect to the signal source. Consider the following scheme:

1. Alice sends a superluminal message to Charlie, who's at rest near Alpha Centauri.
2. Charlie sends an ordinary radio message to Bob, at the exact instant when Bob is zooming by at 0.98c.
3. Charlie sends a superluminal reply toward Earth.
4. David, who's near Earth on a rocket also traveling 0.98c along the line from Earth to Alpha Centauri, receives the message.
5. David relays the message to Bob via ordinary radio.

The exact same calculations apply, despite the extra steps, and the causality paradox persists.

This is because, as I've been emphasizing, special relativity is not a theory of "observations" but rather one describing the geometry of spacetime itself. As such, it doesn't care how the tasks are apportioned between different actors, etc -- what matters is what path the information takes through spacetime. If there's no information (like if the path was traced by a laser pointer), the geometry is exactly identical, including backwards in time steps. But allowing actual information-carrying objects to traverse this path lets me send myself the lotto numbers or something, and that's ultimately where a "strange result" turns into a "causality violation".
Yes, I agree now that SR describes the geometry of spacetime itself. My knowledge had gotten a little rusty...

SR teaches that every time something travels through space in one inertial system, it inevitably also travels through time in another, relatively moving, inertial system. The way it travels through time is described by the Lorentz transformation.

In all these superluminal causality violating thought experiments, the problem arises when a superluminal signal travels through space in one inertial system which leads to traveling through time in another inertial system according to the Lorentz transformation. For instance in the Bob & Alice case, Alice's return signal travels through space in Alice's inertial system which means it travels through time in Bob's inertial system. Something similar happens in your example above in step 3 and 4, between Charlie's and David's inertial systems.

So, the underlying assumption is that superluminal signals behave exactly like subluminal signals: They basically travel backwards in time according to the Lorentz transformation in one inertial system if they travel through space in another inertial system. For me, there is no convincing reason why the Lorentz transformation would be applicable here. We don't have any data on superluminal signals, and don't even know if they exist.

So stating as scientific fact that superluminal travel leads to causality violations is being on thin ice. The scientific evidence for such a statement is non-existing and it is based on the assumption that superluminal signals behave like subluminal signals as described by the Lorentz transformation, while the Lorentz transformation itself does not make sense for v>c.

Has the discussion of superluminal travel, Lorentz transformations and causality developed sufficiently into it's own thing to deserve it's own thread, distinct from a discussion of UFOs and skepticism?

Has the discussion of superluminal travel, Lorentz transformations and causality developed sufficiently into it's own thing to deserve it's own thread, distinct from a discussion of UFOs and skepticism?
Though I enjoyed the journey, the pondering, and being challenged to refresh my knowledge, it did eat away much time from my other duties. So I'll leave it at this. But maybe others want to comtinue the discussion, in which case it does deserve its own thread.

All this talk of FTL - let’s just assume physics holds true and we also want to explain the ufo problem on earth. How can we do that with instances being reported at speeds physically impossible for a human to endure, but still within the bounds of the laws of reality?

[off topic bits removed]

As far as actual aliens getting to us - dunning is correct. The time involved coupled with the short window that we are alive and technologically advanced is infinitesimally small. Lower than abiogenesis.

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All this talk of FTL - let’s just assume physics holds true and we also want to explain the ufo problem on earth. How can we do that with instances being reported at speeds physically impossible for a human to endure, but still within the bounds of the laws of reality?
The survival of occupants makes a lot of assumptions about both (possibly alien) biology and technology, about which we don't have enough info to make a judgment. But if physics holds true, and we have no evidence to the contrary, we have expectations about interactions that should occur, such as sonic booms and burning up in earth's atmosphere. So either all the laws of physics are suspended, or the reported high speeds of some sightings are simply incorrect ...and the latter is more probable, by far.

As far as actual aliens getting to us - dunning is correct. The time involved coupled with the short window that we are alive and technologically advanced is infinitesimally small. Lower than abiogenesis.

Not really. 100 years in 14 billion years is only 1 in 140 million, and it wouldn't even be 14 billion years, as they'd need time to evolve from raw materials on planets that gravitationally accreted too, both notoriously slow processes. And we don't need to be technologically advanced in order to notice alien visitors, so the 100 is an underestimate too. We're well into winning-the-lottery probabilities, probabilities we're capable of agreeing aren't "Infinitesimally small". (Aside: there's a mathematically correct, and simpler, way of describing infinitesimally small probabilities - and that's "zero probabilty". WIth continuous distributions, zero probability things happen all the time, there's no conflict or contradiction.)

The survival of occupants makes a lot of assumptions about both (possibly alien) biology and technology, about which we don't have enough info to make a judgment. But if physics holds true, and we have no evidence to the contrary, we have expectations about interactions that should occur, such as sonic booms and burning up in earth's atmosphere. So either all the laws of physics are suspended, or the reported high speeds of some sightings are simply incorrect ...and the latter is more probable, by far.

Avoiding senescence doesn't seem like a great hurdle, clonal species manage it reasonably well. And even if you can't avoid senescence, acting as a colonial organism, such as The Borg, also works (again, it looks like being clonal is the winning strategy). I'd say the arguments against inter-stellar travel that are strongest are simply the economic ones. Why the heck would an intelligent species waste millions of blob-years (yes, I'm assuming their orbit as well as their structure) with little hope of ever discovering anything worthwhile, it just doesn't pass a cost-benefit sniff test. Yes, I'm committing the fallacy of incredulity here, I accept that.

But yes, once they've arrived at a planet that has observers, they're still going to have to be compatible with Air 1.0, as that's what comes preinstalled. That's the thing that dooms every single one of the more exciting, if you're easily excitable, UFO reports.

Yes, I agree now that SR describes the geometry of spacetime itself. My knowledge had gotten a little rusty...

SR teaches that every time something travels through space in one inertial system, it inevitably also travels through time in another, relatively moving, inertial system. The way it travels through time is described by the Lorentz transformation.

In all these superluminal causality violating thought experiments, the problem arises when a superluminal signal travels through space in one inertial system which leads to traveling through time in another inertial system according to the Lorentz transformation. For instance in the Bob & Alice case, Alice's return signal travels through space in Alice's inertial system which means it travels through time in Bob's inertial system. Something similar happens in your example above in step 3 and 4, between Charlie's and David's inertial systems.

So, the underlying assumption is that superluminal signals behave exactly like subluminal signals: They basically travel backwards in time according to the Lorentz transformation in one inertial system if they travel through space in another inertial system. For me, there is no convincing reason why the Lorentz transformation would be applicable here. We don't have any data on superluminal signals, and don't even know if they exist.

So stating as scientific fact that superluminal travel leads to causality violations is being on thin ice. The scientific evidence for such a statement is non-existing and it is based on the assumption that superluminal signals behave like subluminal signals as described by the Lorentz transformation, while the Lorentz transformation itself does not make sense for v>c.
One last remark: As soon as someone in an inertial coordinate system sends anything with a certain speed to someone else, the thing being sent will exist in its own inertial coordinate system. It can no longer be considered as part of the inertial coordinate system of the sender, since it will have a speed relative to the sender. If the thing being sent happens to be an EM wave, then v=c and hence gamma=1 and t'=t in the Lorentz transformation to any other inertial system.

The relationship between the inertial system of the "thing being sent" and the other inertial systems is given by the Lorentz transformation, which only yields sensible results when the "thing being sent" has a speed smaller than or equal to c. So there is no way to calculate what happens with the Lorentz transformation if the "thing being sent" has superluminal speed.

But if physics holds true, and we have no evidence to the contrary, we have expectations about interactions that should occur, such as sonic booms and burning up in earth's atmosphere.
What if we would have observations of objects which don't burn up and which do not produce sonic booms while travelling at hypersonic speeds? Maybe we'll have to reconsider our understanding of physics then, instead of simply writing them off.
This is exactly what Paul R Hill did.

Paul R Hill was a mid–twentieth-century American aerodynamicist and a leading research and development engineer and manager for NASA and its predecessor, NACA, between 1939 and 1970, retiring as Associate Chief, Applied Materials and Physics Division at the NASA Langley Research Center.
He uses the physics of aerodynamics (which he knew very well) to mathematically proof the absence of heating and a sonic boom if a hypothetical craft can manipulate the surrounding atmosphere using a field akin to gravity (but repelling instead of attracting mass). He spends about 10 pages in the appendices of his book "Unconventional Flying Objects" on the mathematical proof that there is no need for heating or sonic booms in such case.
So, an experimental observation of such craft may be an indication there is more physics for us to discover in the future. Instead of turning a blind eye these cases may be worth investigating.

We, as humans, are already researching similar aerodynamic 'tricks' as the ones Paul R Hill described, using a layer of ionized air (a plasma) around our jets and rockets. Like in this 23 year old research article, which states:
External Quote:
The investigations showed that the plasma or hot-gas injection can be used to reduce drag at subsonic, transonic, and supersonic Mach numbers, resulting in twice (or more) drag reduction. The calculations are in general agreement with the experimental data. The amount of drag reduction depends on jet stagnation temperature.
Source: https://www.researchgate.net/public...rag_Reduction_by_Plasma_and_Hot-Gas_Injection

Needless to say that hypersonic flight, and hence this kind of research, is mostly classified.

So don't be too quick to pose that the absence of heating and a sonic boom point to an observational error.

Avoiding senescence doesn't seem like a great hurdle, clonal species manage it reasonably well. And even if you can't avoid senescence, acting as a colonial organism, such as The Borg, also works (again, it looks like being clonal is the winning strategy). I'd say the arguments against inter-stellar travel that are strongest are simply the economic ones. Why the heck would an intelligent species waste millions of blob-years (yes, I'm assuming their orbit as well as their structure) with little hope of ever discovering anything worthwhile, it just doesn't pass a cost-benefit sniff test. Yes, I'm committing the fallacy of incredulity here, I accept that.

But yes, once they've arrived at a planet that has observers, they're still going to have to be compatible with Air 1.0, as that's what comes preinstalled. That's the thing that dooms every single one of the more exciting, if you're easily excitable, UFO reports.
Consider the same mechanism on a much smaller scale: Earth.
I took a lot of time for the first humanoids to arise from evolution but once they started spreading over the Earth's surface, the 'colonization' of Earth by Homo Sapiens went very quickly, measured on an archeological time scale. It wasn't intentional, either. It just happened because of our natural curiosity and search for new resources and new real-estate.

Fermi calculated what a similar scenario would mean on a galactic scale. And he discovered it would take in the order of 10^6 - 10^8 years for an expanding species to 'colonize' the entire galaxy. They do not even have to travel close to light speed, travelling at a fraction of light speed is sufficient. That is why he came up with his famous Fermi paradox: Where are they?
Enclosed is an interesting article discussing galactic colonization scenarios. It contains a table with several studies done earlier:

The authors of the article developed a simulation program and computed some additional scenarios. They all more or less confirm the Fermi paradox: We should be able to detect an expanding alien civilization by now. Maybe we do occasionaly, but we don't want to face it...

#### Attachments

He uses the physics of aerodynamics (which he knew very well) to mathematically proof the absence of heating and a sonic boom if a hypothetical craft can manipulate the surrounding atmosphere using a field akin to gravity (but repelling instead of attracting mass).
First find some anti-gravity for Hill's hypothetical craft and then we'll talk. You can do a lot of things with mathematics that do not correspond with the known realities of our world, but it seems to be a major stretch to pose (OK, I'll say it) "magic" for items in the low-information zone. The best you can say about Hill's hypothetical is that we haven't proved it to be impossible, but the same could be said for poltergeists and pink unicorns.

Fermi calculated what a similar scenario would mean on a galactic scale. And he discovered it would take in the order of 10^6 - 10^8 years for an expanding species to 'colonize' the entire galaxy.
Similarly, it would take that amount of time for another species to "colonize" us, wouldn't it? ...and it would take a good many other assumptions as well.

Humans spread on a single planet with known habitability: same air to breathe, water to drink, plants or animals to eat, etc, and where those things are not found, neither are people. Although many wild animals would certainly eat us, no known group of them have mounted a concerted war with humans, but we do not now know about any previous occupants of the putative "colonies". And humans had enough backup members right here on earth, far beyond what could reasonably be expected to occupy even a large fleet of spacecraft. Does the earth even have sufficient resources in people and material to build them and energy to power a great number of them? I suggest that realistically any interstellar travel from earth would be extremely limited by necessity.

Yes, technology advances, but the resources of the earth itself, being all we have to work with, are not limitless.

First find some anti-gravity for Hill's hypothetical craft and then we'll talk.
Just look up. Dark Energy is considered a repulsive force. We don't know what causes it, but some species smarter than us, or that already exists longer than us, or a combination thereof, might.

So, the underlying assumption is that superluminal signals behave exactly like subluminal signals: They basically travel backwards in time according to the Lorentz transformation in one inertial system if they travel through space in another inertial system. For me, there is no convincing reason why the Lorentz transformation would be applicable here. We don't have any data on superluminal signals, and don't even know if they exist.
Since the argument doesn't deal in "signals" at all, but only events, if that were to happen the theory would be quite simply contradictory. One set of observers would see one thing, another would see something irreconcilably different: into the trash it goes.

You could hope to avoid the paradox by declaring that not all reference frames are equal, for example, that there's a "superluminiferous aether", a medium with some well-defined rest frame with respect to which FTL signals travel with some speed > c. Which is to say, to assume that relativity is wrong. Hence the dictum: relativity, causality, FTL, pick at most two.

It is important to realize what is being proposed when entertaining the notion of causal FTL travel, which is to discard one of the most solid and well-understood cornerstones of modern physics, without any evidentiary basis.
So stating as scientific fact that superluminal travel leads to causality violations is being on thin ice. The scientific evidence for such a statement is non-existing and it is based on the assumption that superluminal signals behave like subluminal signals as described by the Lorentz transformation, while the Lorentz transformation itself does not make sense for v>c.
The Lorentz transformation doesn't need to make sense for v>c. We use v<c throughout.
One last remark: As soon as someone in an inertial coordinate system sends anything with a certain speed to someone else, the thing being sent will exist in its own inertial coordinate system.
The "inertial coordinate system" is an abstraction. It doesn't exist in reality; it's just a descriptive tools humans came up with.
It can no longer be considered as part of the inertial coordinate system of the sender, since it will have a speed relative to the sender.
Accordingly, concepts such as "being part of a given coordinate system" have no meaning. I can describe objects in a given coordinate system; they aren't a part of them.
If the thing being sent happens to be an EM wave, then v=c and hence gamma=1 and t'=t in the Lorentz transformation to any other inertial system.
I'm assuming that's a typo, since v=c does not give a valid value for gamma.
The relationship between the inertial system of the "thing being sent" and the other inertial systems is given by the Lorentz transformation, which only yields sensible results when the "thing being sent" has a speed smaller than or equal to c. So there is no way to calculate what happens with the Lorentz transformation if the "thing being sent" has superluminal speed.
That is incorrect. The Lorentz transformation can be applied regardless of the speed of the thing being sent because the thing being sent is not even part of the argument. We are describing spacetime itself, and the only thing that matters are the spacetime coordinates, that is, locations and times. "The thing being sent" could teleport to its destination for all we care. As long as it gets to its destination faster than a light beam could, we can evince the causality paradox.

Does the earth even have sufficient resources in people and material to build them and energy to power a great number of them? I suggest that realistically any interstellar travel from earth would be extremely limited by necessity.

Yes, technology advances, but the resources of the earth itself, being all we have to work with, are not limitless.
Fermi envisioned a scenario where only a few large ships are sent to nearby star systems. There, they colonize habitable planets and after about 4000 years these colonized planets, in turn, send a few ships, etc. This causes a snowball effect with an ever increasing rate of colonization. So a single planet only needs to provide the resources for a few ships.
Of course you can think of other scenarios that have the same snowball effect, such as self-replicating probes.

The relationship between the inertial system of the "thing being sent" and the other inertial systems is given by the Lorentz transformation, which only yields sensible results when the "thing being sent" has a speed smaller than or equal to c. So there is no way to calculate what happens with the Lorentz transformation if the "thing being sent" has superluminal speed.

The Lorentz transformation is just a calculational tool between different geometrical systems used to visualize spacetime. You're treating it as describing a real physical relationship which it veritably doesn't.

The Lorentz transformation is just a calculational tool between different geometrical systems used to visualize spacetime. You're treating it as describing a real physical relationship which it veritably doesn't.
I understand what you mean, but I don't understand how this changes the situation. If two objects have a mutual relative constant velocity with respect to each other, the Lorentz transformation applies.

The carrier for superluminal communication can be a tiny container driven by a hypothetical Alcubierre warp drive. Its velocity relative to the sender would be 7.884.000c. Since it is an object that has a velocity relative to both the sender and the receiver of the message, the Lorentz transformation applies. Alas, this transformation does not yield sensible results for v>c, so we're left in the dark. I do not agree with Markus that "we use v<c throughout".

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