FASTER THEN LIGHT (FTL)
(how to break the speed of light without breaking the law)
Just about everyone who's studied the subject knows that special relativity forbids the acceleration of a particle up to and beyond lightspeed relative to an observer. This is supposed to be because the inertial mass of the particle increases with speed, with the increase becoming significant as the speed becomes "relativistic". As the object's speed approaches c (says the argument), the mass of the object tends to infinity, so it would take an infinite amount of energy to take a particle all the way up to lightspeed. Actually, when you look at the initial postulates of SR, they pretty much presuppose that FTL travel doesn't happen.
That's the argument, anyway.
But because SR introduces so many redefinitions, it is actually possible to break the famous "lightspeed barrier" in Newtonian terms without explicitly breaking any SR laws, and to invoke gravitational or acceleration "distortion" effects as an explanation of how a hypothetical FTL object needn't overtake its own wavefront.
EASY STUFF:
(1): Playing galactic hopscotch, and travelling between multiple inertial frames provides a way of achieving (Newtonian) FTL with respect to your startpoint. When multiple intermediate frames are involved, SR helpfully responds by reinterpreting (Newtonian) FTL speeds as being subluminal. To generate those intermediate frames, all you have to do is accelerate (the SR derivation didn't claim validity for accelerating frames, remember?).
(2): "Newtonian" FTL ... - if we didn't know any better, and just slammed our foot down on the accelerator and broke lightspeed, would the folks back home ever know about it? Apparently not, because the "naughty bit" of our journey would again be obscured behind an event horizon. So does (observerspace) SR just describe a lightspeed barrier to observability, rather than what really happens? Is the lightspeed barrier purely an "observerspace" thing? Of course, under GR, acceleration is equivalent to the effect of a gravitational field ...
(3): Gravitational freefall into a black hole - is an example of "legal" FTL travel. However, (a) SR ignores gravity, and (b) the FTL part of a particle's fall allegedly can't be seen by a distant observer anyway. So there's still no direct SR contradiction.
(4): Particle accelerators (on the other hand) are a perfect example of the sort of closed-path observerspace situation that the SR lightspeed limit is likely to be valid for - our last hypothetical example could probably not be brought about using a solar sail and a home-based particle beam. But it's hardly surprising that "dumb" particles never achieve FTL with respect to the coils that are accelerating them, because at recessional lightspeed, you wouldn't necessarily expect the accelerating signal to be able to catch up with the particle to make it go any faster (think about it).
(5): Propulsion systems - some types are definitely subject to an SR-type upper limit. Others may not be.
MORE ADVANCED STUFF:
(6): Time-dilation factors and length-contractions further complicate FTL problems under SR. It seems that some SR effects aren't testable, as a matter of principle, unless you introduce those illegal accelerations again.
(7): Spectral shift arguments under SR predict an infinite forward blueshift for an object travelling at lightspeed, which seems to rule out any higher relative speeds. However, because SR contracts distances (at lightspeed) to zero, this maximum displacement per unit time, when calculated using non-contracted distances, actually corresponds to an infinite speed in Newtonian terms. End of problem - it seems that "speed" under SR isn't necessarily the same as "speed" under Newtonian mechanics.
(8): Probe chain theory - is the logical consequence of any theory that includes velocity addition formulae. The formula says that it's possible to view an object receding with any speed whatsoever, as long as the light passes through enough intermediate inertial frames for the recession velocity between pairs of adjacent frames to be less than c. That means that FTL signaling (in one sense) ought to be possible, because the probe chain provides an accelerating path for the signal. Most physicists won't like this, but it appears to be an inescapable consequence of the SR math - so hey, don't blame me, guys, this is a side effect of your theory. I'm just the bearer of bad news - this situation ought to have been sorted out decades ago. See also the "magic window" problem. * 9: Who said space has to be flat? - As the SR flat-space assumption fails for acceleration and gravitation, and also seems to fail in multi-body inertial systems, and in two-body inertial systems when you consider the gravitational effects of kinetic energy at high speeds, then why should we believe that it holds for any situation where v>0?
The SR lightspeed barrier limits the recession speed that an observer can directly induce in a particle, and it limits the speeds achievable in a single simple explosion, but it doesn't seem to limit the speeds that a particle can accelerate to, provided that the particle itself is providing the energy. FTL might still not be practical - if you get hit in the eye by a bit of grit travelling at 300,000 km/s, it could ruin your whole day - but the current arguments against it seem to evaporate when you stick them under a microscope. __________ ______ ____ __________ ______ ____ __________________
So, we find that the SR lightspeed barrier, in the sense that it is usually portrayed, might not really exist other than in an observerspace projection. It might just be an arbitrary "mapping" artifact with no physical existence beyond an individual observer's situation, like the Earth's horizon.
The "infinite mass" argument is an observer-effect, particle accelerator arguments don't necessarily apply to situations where objects have their own propulsion systems, the SR description is compatible with "deduced" FTL recession hidden behind an event horizon, and the sort of behavior this gives is already accepted by mainstream physics, as happening when something falls into a black hole, and FTL Newtonian approach velocities also seem to map to subluminal velocities under SR, there's no contradiction there, either.
Gravitational and acceleration arguments seem to say that FTL is legal, but are outside SR's jurisdiction, and SR ignores gravitational effects that occur under GR, despite the fact that GR is supposed to reduce to SR for constant-velocity motion. Given that kinetic energy (under GR) has gravitational effects, there's no way you can guarantee that any GR inertial system can be described in a flat-space theory. Reinterpreting Lorentz shift as gravitational destroys the SR propagation model, and means that there are lightspeed gradients between components of inertial systems.
It only appears impossible for a particle to be emitted at >c, because the 959f512j coupling-efficiency of applied radiation drops to zero for c-receding objects. A conventional rocket can't expel gases at FTL speeds (relative to itself), but that doesn't necessarily mean that by continuing to expel gases at subluminal speeds (relative to itself), it can't eventually achieve a speed relative to its start-point that is significantly higher than the speed of the expelled propellant.
In short, the SR lightspeed barrier is based on a flat propagation model that has no convincing evidence in its favor, that fails in almost every situation that it is applied to, and which (if we accept the equivalence principles of GR) can't be correct, anyway.
That's not to say that there might not be other factors that would make faster-than-light travel impractical, but you won't find those other factors by studying SR
So, we might not be able to achieve useful FTL travel after all, but if we do manage it, at least we'll be able to use a probe chain get a message back to the folks at home.
Black hole bungee - jumping, anyone?
'Treason never prospers'; And, aye, there's a reason For if it doth prosper, none dare call it treason ...
old English proverb, Anon __________ ______ ____ __________ ______ ____ __________________
Some of the terms used have slightly different meanings on different pages. The language of physics is still quite crude in a number of ways, and doesn't always have different words for similar things (for instance, most physicists are happy to talk about "the Doppler formula", but don't have words to distinguish between the different variants, which is why I had to make up the eDoppler/oDoppler terminology). I once sat down and tried to see how many definitions there were for "mass", and ended up with about fifty, some of which normally coincided, some of which didn't.
"Lightspeed" is another extremely slippery word.
In these pages, "faster than light" or "FTL" is used in the sense that it us usually used in "spaceship" questions, where what you actually want to know is how quickly one can travel between two agreed coordinates (such as two cities, or planets). In this context, expressing the velocity in terms of a redefined distance isn't particularly helpful, and neither is redefining speeds in terms of the deduced local value of c in the moving observer's frame. I'm not suggesting that a spaceship can overtake its own lightsignal, because I'm not accepting the (unverified) SR assumption that lightspeed is anything other than locally constant.
(1): Galactic Hopscotch
(island - hopping taken to extremes)
Let's suppose that a galaxy is receding from you at 0.6 lightspeed, and a second galaxy is receding from you at 0.6c in the opposite direction. The combined (Newtonian) recession velocity is 1.2c. Can a spaceship leave one galaxy and reach the other?
Let's try it. A spaceship leaves galaxy A at 0.9c and heads blindly in the direction of the other. After a lot of time, it decelerates a bit, and finds itself at the mid-point of the two galaxies, with each receding at the same speed of 0.6c. With the engines off, it is now sitting "stationary", with the two galaxies A and B each receding at 0.6c in opposite directions, just like we were in the first paragraph. So far it hasn't broken lightspeed, and everything is legal.
Now comes the fun bit.
Since it has managed to get to its present position from its home galaxy (A), it must be able to get back again - after all, this only requires it to travel for a long enough time at less than lightspeed.
No problem.
However, there is no real difference between firing up the engines to go home, and expending the same amount of fuel to go on to galaxy B. A and B are now equally reachable. In fact, if the ship's crew spin the ship around enough times, while they try to make up their minds, and forgot to make adequate notes on their outward journey, they might lose track of which galaxy, A or B, they actually came from!
So, let's suppose that by accident or design, the crew end up heading for B. Eventually, they get there. They have now managed to accelerate (in two stages) up to a Newtonian speed of 1.2c with respect to their home planet, again, without breaking any SR laws.
How did they do that?
Well, basically, the problem is one of definitions.
If an object is receding at a certain Newtonian speed, then the (directly - observed) recession speed under SR has a particular value. However, the moment that light from a receding object is passed via a third intermediate frame, SR chooses to reinterpret the total velocity as being less than the sum of the two halves - the "supposed" velocity of the object actually depends on the characteristics of the light-path (see SR velocity-addition formula, magic window problem).
So - an object (or inertial frame) can be receding at up to 2c, and you can still reach it, simply by going via a third intermediate object (or inertial frame), which is used as a sort of staging post. In fact, the test particle can have any recession velocity whatsoever, provided that it uses enough intermediate frames. There's actually no upper limit at all. Any combination of velocities that are each less than c add together (under SR) to a total velocity that is also less than c, so with enough intermediate frames, SR can take any recession velocity whatsoever and redefine it as being conveniently less than lightspeed.
So - what constitutes an intermediate frame?
Well, if you want to send a light-signal, it could be a physical object (see probe chains), but if you were sending a physical object, the intermediate inertial frames could be supplied by the object itself, simply by turning the engine on and off to create inertial "coasting phases" between bursts of acceleration.
The next logical step is to conclude that even the coasting phases are unnecessary - all the ship would have to do is leave the engines on.
Special relativity is based on the assumption of flat space and observed reality. Any object receding at greater than c can't be directly observed within flat space, and therefore can't (normally) exist within the theory. A hypothetical object receding at greater than c couldn't be hit by a ballistic object launched by the observer at less than lightspeed, so (since an object can't be accelerated by the observer to greater than c), the logical assumption is that speeds >c aren't achievable.
The critical weaknesses in this argument concerns acceleration. Einstein's SR is purely "ballistic", and deals with constant velocity intervals between pairs of inertial frames. It Doesn't Do Acceleration (remember this, it's critical).
If you wanted to be shot out of the mouth of a Jules Verne -type cannon at >c then you'd probably be disappointed, but if you were launched in a rocket at less than c, and kept accelerating then special relativity would obligingly keep reinterpreting your final velocity as being less than c, whatever its value in Newtonian terms.
The velocity-addition formula is SR's greatest get-out clause.
Just put your foot on the accelerator and keep it there.
Notes: the SR lightspeed limit does not hold across multiple frames.
"Ballistic" arguments have traditionally been a weak point for physics theories. The aeronautical equivalent would be saying that it is impossible to construct a simple craft capable of flying from London to New York, because in order to complete the journey, the initial speed of the craft (when launched from a gun) would have to be so fast that it would burn up in the atmosphere. You can get around the argument by deciding to continually apply thrust, instead of expending all your energy in one initial blast.
Another ballistic argument once "proved" that it was impossible to launch a payload into space with a liquid-fuelled rocket, because the fuel tanks, etc., would always be too heavy to get the rocket all the way up. That line of reasoning evaporated when someone thought of launching rockets from the top of other rockets (multi-stage launchers like the Saturn V).
(2): What happens if you try it?
(in which a craft breaks the lightspeed barrier,
and the folks back home are none the wiser)
The last section showed how the SR velocity-addition formula allows an object to achieve a Newtonian recession speed greater than c relative to its start position, provided that it undergoes acceleration and that the accelerative force doesn't originate from its startpoint.
How is this explained? Quite simply. According to SR, you can't just add two velocities together using standard addition, you have to use a special velocity addition formula. When an object recedes at 0.5c, SR says that its directly - observed frequency is multiplied by a factor of 0.577... If it recedes at lightspeed, the frequency drops to zero. But if a signal is fed through two stages, each with a recession velocity of 0.5c, then the observed frequency of the furthest object (receding at lightspeed) only drops to (0.577..)², or 0.3333', rather than zero (which you'd normally expect for lightspeed recession under SR). The standard SR rules simply don't work the same way when intermediate frames are concerned - an object with a particular supposed recession speed could be "legal" or "illegal", simply depending on whether or not you looked at it through a moving sheet of glass (the "magic window" argument). That's difficult to reconcile with SR's model of simple flat space whose light-carrying properties are unaffected by the presence of objects with constant relative motion - it implies that there is a much more subtle set of effects in action than the simple SR lightspeed limit would have us believe.
The SR velocity-addition formula simply takes the way that composite shifts deviate from the SR shift law, and works backwards to generate a new velocity value that would generate the same result, were the object being directly observed, instead. When you multiply two f'/f shifts that are each greater than zero, the result is (obviously) still greater than zero. So, under SR, when you multiply to shifts each caused by a recession velocity less than c, the combined f'/f shift also never reaches zero (as it would be expected to at recessional lightspeed), and the resulting SR velocity is therefore deemed to be less than lightspeed as a matter of principle. Where the SR velocity/shift law fails for composite shifts, SR simply redefines the velocity from the shift value so that it doesn't fail.
This remapping is, of course, totally arbitrary, and lets us bring any superluminal velocity down to less than c without changing what the object is really doing. It's a way of imposing a lightspeed barrier on paper by redefining any composite shift to be less than c, provided that a signal can cross the intervening distance, courtesy of an intervening "carrier frame" (see probe chains). So although SR says that nothing can travel faster than lightspeed, it also says that every velocity greater than c can be re-interpreted as being less than c, provided that there is are enough intermediate frames for a signal to pass through.
That's why that galaxy-hopping example worked. We know that the total velocity is 1.2c, but SR decides to call it 0.88..c instead, so that it doesn't break the notional SR lightspeed limit when the ship crosses from frame A to frame B. The ship's own 2-stage acceleration provides the intermediate frame necessary to justify the application of the velocity-addition formula. (NB: this value was calculated under SR from two instantaneous velocity-changes each of 0.6c, the total figure reduces further when more intermediate frames are involved)
So, is the SR c-barrier a pointless fiction?
Not quite. To understand what's going on, you have to remember that SR is a theory that describes observerspace, that is, it describes phenomena as they appear to happen. The addition of the extra frames does make the object seem to be receding at a slower speed, if speed is judged from the SR shift formula, so under SR, that speed reduction is deemed to be genuine. Now, if we suppose that the spaceship in our last example accelerates smoothly up to 1.2c without taking a mid-flight break, then what would the back-home observer see? Let's try an alternative interpretation of the observed phenomena:
- They see the ship to be progressively time-dilated as it approached recession lightspeed, and as the ship is seen to approach the point in space at which c-recession would actually be achieved (the "c-point"), the ship's clock appears to be ticking so slowly that it effectively seems to stop. In our hypothetical situation, the ship would actually whizz pass this space-coordinate at midnight, ship-time, an indicator light would come on in the ship's cabin, and the crew would have a bloody good party to celebrate. Back home, though, the ship would never be seen to pass the c-point coordinate, and would eventually seem to be hanging in space, ridiculously redshifted, always gaining on the coordinate but never quite reaching it, the hands on the clock never quite reaching the twelve o'clock position. The party would never be seen to happen, and the home astronomers might sorrowfully conclude that the unfortunate traveler's fate was to be frozen there until the end of time.
Nothing that happened to the ship after it crossed the-point would ever be seen to happen, back home.
"See?", one of the astronomers might shout angrily at the others, "I told you that FTL travel was impossible!" __________ ______ ____ __________ ______ ____ __________________
Conclusions:
Because SR is an observerspace theory, what it actually says is that lightspeed recession can never be observed to happen, i.e., that this phenomenon can never happen within observerspace. Our example described the ship happily accelerating beyond c, but everything that happened after the ship reached c-recession was shielded from the home observer by an event horizon, so the home astronomers saw exactly the same things that they expected under SR, even though something else was "really" happening. The fact that SR forbids FTL recession isn't automatically evidence that such behavior can't happen. The observed classes of effects are the same, either way.
This talk of event horizons may have raised some eyebrows. Aren't event horizons strictly a "black hole" thing, i.e. a gravitational effect?
Yes and no. Read on ... __________ ______ ____ __________ ______ ____ __________________
Notes: the SR lightspeed barrier fails for situations involving accelerations.
The observed-clock-stopping of the ship assumes, of course, that the ship isn't being observed through any intermediate frames. If the ship's crew threw a series of champagne-glasses out of the window as it accelerated, then if you looked at the ship through this expanding trail of garbage (or through the ship's exhaust plume), then the same SR velocity-addition law that made us suspect that this hypothetical FTL behavior was possible in the first place would also conspire to make the ship visible beyond the c-point, and to redefine this "visible" velocity to less than c. Which would mean that the ship's behavior wasn't illegal after all (!).
It turns out that the ship's recession velocity is only illegal if you can't see it, and if you can't see it, then it isn't dealt with by SR. It seems that whenever you try to break the SR law, SR either ignores your existence or forgives you completely.
(3): Using gravitational gradients to break lightspeed
(in which a ship's crew decides to jump into a black hole, just for fun)
Using existing gravitational gradients.
In the last section, we described the observed effects of a hypothetical object smoothly accelerating up to a speed greater than c, and appearing (to the home observers) never to actually reach lightspeed. Although we showed that this sort of acceleration could be consistent with the observed effects predicted by SR, we didn't make any attempt to explain how the acceleration was achieved. It certainly couldn't be achieved by the use of a solar sail and a home-based laser, because this acceleration would still suffer a decrease in efficiency at relativistically - significant velocities.
So - is there any known force that might be capable of generating this sort of acceleration?
Yup It's called gravity.
Let's repeat the last hypothetical experiment, but this time, the ship isn't powered by some "magical" engine, but is simply falling directly away from the observers, into a black hole.
As the ship drifts towards the hole, it picks up speed, slowly at first, and then as it comes more under the influence of the hole's gravitational field, more and more quickly. The ship undergoes a smooth acceleration away from the observer, as before, and once again, the home observers see the ship to be progressively more time-dilated as it accelerates (and falls into the hole's gravitational well).
Again, the ship's journey has been calculated so that the recession velocity ought to hit lightspeed at exactly midnight (ship's time), and the relevant space-coordinate where this is expected to happen has been marked out on the map. This c-point is the point at which the escape velocity of the gravitational field equals that of light - in other words, it lies at a distance from the centre of the hole corresponding to the hole's Schwarzchild radius - the c-point is at the hole's event horizon.
Again, the (not terribly bright) crew celebrate the chiming of midnight on the ship's chronometer with the popping of champagne corks, and a radio message broadcast back home, "We did it!".
Again, the home-based astronomers monitor the ship approaching the c-point, and see the ship's clock grinding to a halt as its hands approach the twelve o'clock position. Again, they never see the wild party, they never get the jubilant message, and they never see the craft ever reaching the coordinates that signal its achievement of recessional lightspeed. The craft appears frozen in time and space, hovering impossibly at the event horizon, with its signal (corresponding to the last few seconds before midnight), stretched out to infinity.
In both the "gravitational" and "powered acceleration" cases, the ship is never seen to reach lightspeed. However, in the "gravitational" case, this observed effect (of the ship hanging forever at the edge of the black hole event horizon) is considered to be an artificial illusion. Nobody (says the current prevailing wisdom) would actually be dumb enough to believe that the craft is really stuck at the event horizon, would they? Surely anyone with a brain would realize that there's no reason for the craft not to continue its fall, and that the stopped clock is simply an observer-related effect caused by the properties of signals leaving the craft?
Perhaps. However, the treatment of the observed clock-stopping as "real" in a way that goes beyond the observerspace definition of the word, is exactly how most SR people treat the first example. __________ ______ ____ __________ ______ ____ __________________
Conclusions:
The first "fantasy" example of a ship accelerating up to and beyond lightspeed has a near equivalent in the behavior of an object free falling into a black hole. In both cases, an event horizon prevents the observer from seeing any hypothetical FTL behavior that would be expected to occur beyond the c-point.
In the case of a black hole, the unseen FTL behavior is considered to be a genuine effect, and the lightspeed barrier is considered to be an illusory by-product of the event horizon (the mathematical surface around a black hole at which infalling matter is expected to achieve lightspeed relative to a distant outside universe).
In the case of "powered acceleration", the same behavior is interpreted differently, the clock-stopping is considered to be "real", and the acceleration to superluminal velocities is therefore deemed "really" not to happen. However, there seems to be no real reason why the same "black hole" logic can't also be applied to the SR lightspeed barrier.
But surely we know that the lightspeed barrier is genuine where gravity is not a factor? Don't particle accelerator experiments show this?
This is dealt with in the next section ... __________ ______ ____ __________ ______ ____ __________________
The black hole example is an obvious case of objects having FTL velocities in modern cosmology. Another is the case of Hubble recession, where the "natural" recession speed of objects increases with distance, and has no upper limit other than the extent of the universe (this is the reason why the two galaxies in the "galaxy-hopping" example were allowed to have FTL recession velocities in the first place). Both examples involve the distortion of space, either through explicit gravitation (black hole) or large-scale curvature (Hubble recession&shift), so the SR "flat-space" model is incapable of dealing with them, just as it wasn't able to properly cope with the acceleration example (acceleration warps space, too). According to the (non-standard) DMS model, even constant-velocity motion warps space, with the Lorentz correction under SR producing an approximation of the effect. And since SR predicts the kinetic energy of an object becoming significant at relativistic speeds, then according to GR, this energy has a gravitational effect. If the effect is the Lorentz effect, then the SR flatspace model is wrong, and there's velocity-dependent curvature (gravitational effects). If it isn't the Lorentz effect, then the SR shift formula just isn't accurate at speeds approaching c. It's a good rule of thumb that whenever you come across an example where you'd expect FTL behavior to occur, SR has a good excuse to stop working.
Under gravitational freefall, the ship can achieve FTL with respect to an external observer, but it doesn't overtake its own lightsignal.
(4): Particle accelerator behavior
(if something runs away faster
than a speeding bullet,
you can't shoot it)
The SR argument for mass-dilation in particle accelerators seems to go something like this:
In the frame of the particle accelerator, the accelerating signal travels at c, so it can't catch up with a particle that is already receding at lightspeed. If the particle's resistance to the force of the beam is taken as a measure of its inertial mass, then the fact that the particle is unaffected by the beam would tend to lead to the conclusion that the particle has infinite mass.
In the frame of the particle, the beam is also supposed to travel at c, so it does catch up with the particle (under SR, the beam is always supposed to travel with fixed speed relative to any given observer). This apparently-paradoxical result actually generates the same final result, because the particle (under SR) will see the accelerator coils to be receding at lightspeed, and therefore to be "clock stopped" (through inertial mass dilation), so the particle again has zero interaction with the beam, but this time the zero energy of interaction is reasoned to be caused by the "zero" frequency of the current in the coil, rather than because of propagation factors.
So, under SR, the deduced behavior of the beam is different for different observers, but the final observed effects are consistent in each case. This sleight-of-hand is both a weakness and a strength of SR - it provides a way of allowing any observer to reason that lightspeed is everywhere constant with reference to their own frame, at the expense of postulating an additional shift mechanism that is supposed to be separate from propagation factors as a matter of principle, but whose separate existence is (by the same principle) always going to be experimentally unverifiable if the theory is correct.
As the accelerator result can be arrived at by either a Lorentz shift or by a purely propagation effect, or indeed by a range of intermediate solutions, or by a propagation effect that includes the Lorentz shift as a gravitational distortion effect, this behavior can't be taken as ironclad evidence that SR's predictions in other related areas are necessarily going to be correct when taken out of their (flat-space, observerspace) context. The DMS model discussed elsewhere on this site (for instance) also generates a similar effect in particle accelerators, but doesn't preclude FTL travel in the wider sense of the word. A similar situation occurs with a range of gravitational theories.
In other words, SR supports one unverifiable assumption (global lightspeed constancy for all observers) by introducing a second (separation of motion shifts into propagation and Lorentz components), and then hedges those assumptions with a range of other techniques (like clock synchronization methods) whose purpose is to make sure that certain other interlinked properties of the theory are also unverifiable. __________ ______ ____ __________ ______ ____ __________________
Conclusions
Particle accelerators do provide a special case situation where the "infinite inertial mass at recessional lightspeed" argument appears to hold. However, this effect can be calculated from straightforward propagation arguments, i.e. you wouldn't necessarily expect an EM pulse to be able to transfer energy to a particle that it wasn't supposed to be able to catch up with, so the observed mass of the particle when receding at lightspeed would normally be expected to appear to be infinite in this class of test, in a wide range of models, simply because of propagation issues. In the case of force applied by the observer, you'd expect observerspace arguments to hold.
In other cases, the effect is less clear-cut. You can't just extrapolate the particle accelerator results to different classes of experiment, and assume that you'll get the same behavior.
Those particle accelerator tests don't tell us whether this upper limit also applies to situations where it isn't the observer that is supplying the accelerating force, e.g. particle anchored to distant object by long rubber band, particle fitted with own propulsion unit, particle attracted to distant gravitational body. In the last of these cases (gravitational acceleration), we don't believe that the SR limit holds, and GR's equivalence principle between gravitational and non-gravitational acceleration effects rather implies that this breakdown may bleed through to the other "acceleration" cases, too (there may be other complicating factors).
We have little evidence that the c-limit is a practical upper bound in anything other than purely observerspace situations (force applied from observer's frame), and we still don't know whether the lightspeed upper limit applies to craft with simple rocket motors.
In order to evaluate whether the inertial mass is increasing in a wider sense, resistance to applied acceleration isn't terribly useful. Resistance to applied relative deceleration (i.e. electromagnetic braking efficiency) might be a more meaningful test. I haven't heard of anyone carrying out this sort of test, but I haven't (yet) looked terribly hard. Rumblings from particle accelerator people that SR mass-dilation is an "old-fashioned" idea are interesting - it implies that there may be classes of particle accelerator tests that produce non-SR results in this area. Could this data deal with blueshift/deceleration stuff, the region where SR and Doppler diverge? Will try to get someone to talk ...
(5): Propulsion systems
Hypothetical "inertial drives" (aka "gravity drives"), which work by generating a gravitational gradient in a particular direction, and allowing the craft to "free-fall" in that direction, are allegedly not subject to a lightspeed limit under GR. Unfortunately, most of them depend on "exotic matter", which we have no reason to believe exists. Inertial drives based on DST's may also be possible, but are probably subject to the same high-speed efficiency drop that affects particle accelerators. Conventional propulsion units don't seem to be affected.
a) "Exotic matter" inertial drives:
"Exotic matter" drives are dealt with rather well in "The Renaissance of General Relativity" by Clifford M.Will (Published in "The New Physics", ed. Paul Davies). Exotic matter is a purely-hypothetical form of matter with negative gravity (there are reasons to suppose that it doesn't exist). What is interesting about the subject (in this context) is that a hypothetical gravity drive constructed from a combination of "exotic" and "normal" matter would (allegedly, under GR) not be affected by the lightspeed limit (M. Alcubierre published another variation on the ematter drive a couple of years ago). I'm really not convinced about this "exotic matter" stuff, but there are other possible ways of constructing inertial drives, and if GR officially puts no upper limit on the speeds attainable by an ematter drive, there's a faint chance that this might also apply to the other categories.
b) DST inertial drives:
DST's will be dealt with in a future document.
However, on a brief examination it seems that although DST's do seem to have a lot of nice properties, I'd guess that an ability to avoid "particle accelerator" syndrome isn't likely to be one of them. I doubt that a hypothetical DST engine would be capable of taking you past (or even up to) environmental lightspeed, although it might be useful for braking. I might well be wrong on this one, though.
c) Remote-powered drives
One of the smart ideas that has become popular recently is the idea of building a very small, very low-mass probe, fitting it with a lightweight "solar sail", and then accelerating it up to god-knows-what speed by beaming energy at the sail from a remote site. The BIG advantage of this method is that you effectively leave your engine at home - a probe weighing a few pounds (plus sail) would then be able to get the full benefit of a particle-beam/laser/whatever apparatus weighing possibly several hundred tons, with having to bring it along. Not only is (almost) all the utilizable power going into accelerating the payload (plus sail), but you get to keep the engine! Thus a single "driver" could be situated at a handy site, close to its fuel supply, and reused over and over again on a succession of lightweight probes.
Unfortunately, this (or any other remote-powered drive system) would seem to be limited by the lightspeed barrier. <sudden thought> ...unless you stacked a chain of them, and powered the whole chain by a single "fixed" driver? Hmmm...
For the purpose of this exercise, I'm not classing "nuclear bomb" drives (where you throw a nuclear device out of the back of the craft, and ride the shockwave, then repeat the exercise indefinitely) as "remote powered", but as "conventional" (technically, the initial blast is "remote", the rest aren't).
d) Conventional propulsion systems
This one is the big surprise.
It seems that your common-or-garden sacrificial-propellant rocket motor might not necessarily be governed by the SR lightspeed limit.
Let's suppose that you take a lump of matter, divide it into two halves, and set off a nuclear explosion (or something), causing the two halves to fly away from each other at 0.5c. Then you take one of the two pieces (travelling at 0.25c wrt the immediate environment), and repeat the exercise, blowing it apart so that the two pieces (quarters) again recede at 0.5c to each other. One piece is now stationary wrt the immediate environs, the other (according to Newtonian mechanics) is travelling at half lightspeed. Now keep repeating the exercise with the "fast" piece. Under Newtonian laws, this (terribly inefficient) method of propulsion would end up accelerating a tiny part of the total mass up to and beyond lightspeed.
Interestingly, this Newtonian simplification doesn't seem to be quite illegal - the fact that we aren't trying to achieve lightspeed in a single step means that the usual SR prohibition of FTL behavior doesn't hold - all those intermediate steps mean that the final (deduced, superluminal) velocity is helpfully reinterpreted by the SR velocity-addition law as being less than c, provided that the separation velocity of each division is also less than c (which it is). A signal could still be passed along the expanding chain of "propellant" pieces, from end to end, even if the two ends had a Newtonian recession velocity greater than c, so in that sense, the speed of light hasn't been exceeded, and the situation appears SR-legal.
SR prohibits the separation-velocity caused by any single explosion being greater than lightspeed and it appears to predict a loss in propulsive efficiency per explosion, compared to the Newtonian model. It also prohibits the home-based observer from being able to directly observe the lightspeed barrier being broken - to them, each explosion would have a progressively smaller result, until, close to lightspeed, the observed effect of each explosion would be minimal, and the interval between explosion would eventually become so long, that the explosion that would be expected to take the payload past lightspeed would never be seen to happen. But we've already shown that this sort of observed behavior doesn't necessarily mean that the payload wouldn't be happily accelerating past recession lightspeed, unobserved - it could be reinterpreted as the effect of a non-SR light-propagation model on an accelerating object (see black hole example). We just don't know. A second observer, in front of the accelerating payload might see a rather different result. *note
Unlike inertial drives, the characteristics of two pieces immediately prior to an explosion, measured within their initial frame, are supposedly independent of the motion of the background (frame-invariance), and the acceleration forces on each half, and the final speeds ought therefore to be symmetrical about the initial center of the piece. According to current theory, both the payload and the propellant pieces would be similarly mass-dilated before an explosion, which leads us to expect that both pieces would fly off at identical speeds relative to the "rest frame" that they had just before to the explosion. There are some problems with this, because the two pieces would be expected to have different amounts of mass-dilation after the explosion, depending on their relative velocities to the environment, but accepting this seems to lead to a breaking of the principle of the equivalence of inertial frames, so the effect _might_ be another observer-specific thang.
Under Einstein's definitions, the SR model fails when applied to smooth acceleration (except as a rule-of-thumb approximation where the amount of acceleration is minimal), and when the acceleration is taken in a series of quantified steps (e.g. explosions), the SR velocity-addition formula cuts in to make sure that any Newtonian superluminal velocity that _might_ be presumed to occur is quickly redefined away as being subluminal, as a matter of principle. Either way, Newtonian FTL does not seem to be prohibited, except as the result of a _single_ explosion, whose separation speed would then be limited by the speed of radiation in that frame (c). __________ ______ ____ __________ ______ ____ __________________
For an explanation of indirect viewing and the significance of the SR velocity-addition formula's reduction of summed velocities that would normally equal >c to less than that of light, see the section on probe chains.
(6): Time-dilation issues
(in which special relativity manages to predict exactly the same sort of observed t-d effects as Newtonian FTL, but with a different interpretation)
Let's try describing the event horizon argument of section 2 in another way.
A manned spaceship equipped with a rocket pack theoretically capable of taking it up to superluminal speeds according to Newtonian mechanics is launched, and the stay-at-home observer sees the object ageing more and more slowly, and finally grinding almost to a halt at the c-point. How do we tell if the clock slowing is just an illusion?
Let's consider the situation from the pilot's viewpoint.
The "naive" Newtonian prediction: The ship leaves earth and fires up its engines, expending so much energy that it reaches a superluminal speed. It then crosses a distance of one lightyear in less than a year. This is verified by the ship's log, and observations made en-route.
The "modern" SR prediction: The ship leaves earth and fires up its engines, expending enough energy to bring it up to superluminal speeds in a Newtonian model. However, we "know" that mass-dilation becomes significant at high speeds, so we can deduce that the speed never actually reaches c, and the journey therefore takes more than a year. However, the mass-dilation effect also slows the ship's chronometers (and all biological processes) by the same amount, with the result that the journey appears to take less than a year, ship-time. However, the uncomfortable fact is that while SR is supposed to make FTL travel impossible, the pilot onboard the ship simply doesn't see the c-limit operating, and the ship's crew think that they've traveled to their destination at an FTL velocity, as planned.
In the second example, how do we persuade the crew that they are wrong?
This is where SR pulls another trick. Because the shipboard observers see their journey to take so little time that it must have taken place at FTL speeds, and because their own time-dilation can't be considered to be real in their frame, SR alters the only remaining parameter. It alters the length of the journey. "There!", says the onboard theoretician to a skeptical crew, "The journey didn't take less than a year due to our travelling at FTL speeds, or to time-dilation - it took less than a year because the entire outside universe contracted along our direction of motion. So when we passed all those marker beacons at 1000km intervals, they weren't really 1000km apart - they were closer together. And when we saw the entire outside universe -- stars, planets, etc -- whizzing past at more than 300,000 km/s, those weren't _real_ km, because they were moving! It's just that the entire outside universe sneakily conspired to make it look as if we were travelling at more than c - we weren't really. It was all an illusion!"
At this point, the theorist would probably be thrown out of the airlock without a suit.
As the theoretician is bundled into the airlock, he makes one last plea for understanding.
"I can prove it!" If we turn the ship around and head back for home, the fact that we've been moving relative to our environment will mean that our chronometers will show less time to have elapsed than the ones back on earth.
The captain thinks. "But shouldn't _we_ see _their_ chronometers to be slow?"
"No - because they weren't really moving - we were!" *
<The airlock bolt is drawn back.>
"Look, ok, I know that that contradicts SR, but the difference is because of environmental factors - we are interacting differently with our environment"
"You mean, gravitationally? Because I thought gravitation or privileged frames invalidated SR..."
<click>
"No - wait! It's possible to set up a symmetrical experiment with two ships, so that the environmental factors cancel out!"
"And then, does one clock end up slower than the other?"
"Actually no, there's an extra 'relative acceleration' blueshift that cancels out the timelag when the two ships turn round. In a symmetrical test, both sets of clocks agree at the end of the experiment." **
<click>
"And I suppose that extra blueshift isn't compatible with the SR model, but it's deduced to happen every time we accelerate towards something? And there's no way we can head home to check out your claim without accelerating towards it and invalidating SR anyway? So we just have to take your word that this SR effect exists, because you say there's no real way to verify it without violating the SR rules? And the human race abandoned the idea of FTL travel for a century, because people _believed_ this stuff?"
"Er..."
<clikWHOOOOSHHHHHHH>
a) Not a particularly credible argument under SR, which denies the existence of absolute motion. Still, it crops up in at least one textbook on SR.
Apparently,
this "extra SR" acceleration blueshift formula is given in MTW's "Gravitation", even though it's incompatible
with the basic SR model. I haven't been able to check this, though, because
nobody in the UK has the book in stock - it's currently between reprintings (23-03-96).
(7) Spectral shifts
(redshifts aren't a problem,
blueshifts require a bit more head-scratching)
As we've seen clock-stopping at recessional lightspeed is a feature of a range of propagation models, some of which seem to allow FTL behavior. It's not unique to flat-space SR (see event horizons). Irrespective of how much redshift you see in a receding object, SR will always deem the velocity to be less than c. __________ ______ ____ __________ ______ ____ __________________
It is natural to assume that an object approaching at lightspeed rides its own wavefront, and therefore appears to have an infinitely short wavelength. This is a feature of the SR propagation model, and seems at first sight to be an unavoidable consequence of the theory. A further increase in speed would therefore be expected to result in the object overtaking its own light, with the corresponding "tachyonic" difficulties.
However, let's now go back to that last "spaceship" example, where a ship managed to cross a one light-year distance in less than a year of ship's time, without violating SR (although it appeared to the "ignorant" ship's crew that they'd broken the lightspeed barrier). That example was justified (under SR) by a contraction of the spaceship's path, and a redefinition of the ship's speed to less than c. Wouldn't this "Newtonian" FTL lead to the object overtaking its own wavefront, and wouldn't the fact that the ship is approaching its destination at greater than Newtonian c, ship-time, lead to the light that's coming from the destination and hitting the front of the ship being blueshifted to infinity?
Weirdly enough, it doesn't. Although the ship crosses the set distance in less than a year, ship-time, the blueshift of the oncoming radiation isn't infinite, because it is now predicted from the deduced SR distance traversed per unit time, which is, of course less than c due to the notional path-contraction. If "speed" is defined using the ship's time and the environment's rulers, then the object's destination can approach at any speed whatsoever, and (under SR) the blueshift will always be finite. The "SR speed" of the object is always helpfully redefined as being less than lightspeed.
How about the way that the destination sees the approaching ship?
If the above effect is symmetrical (which it should be under SR), then the blueshift seen on the approaching ship again doesn't reach infinity until the ship is again travelling at an "infinite speed", with distance instead measured by the ship's rulers, and time measured in the background frame.
The notion that "moving rulers contract along their direction of motion" would have you believe that the approaching ship's rulers are shorter when it's speed is greater, and that this new notional "infinite speed" therefore occurs when the ship is travelling at Newtonian lightspeed, and that Newtonian lightspeed is therefore the limit at which an approaching ship has an infinite blueshift. Unfortunately for this idea, SR rulers don't just contract along their length when they move, they also contract or lengthen according to whether they are approaching or receding from an observer, by an amount inversely proportional to the Doppler shift on the object (NB: not a lot of people know this).
note 1:
The fact that the usual SR descriptions deal only with the special-case transverse behavior of rulers ought now to be making your stomach curl up into a little knot, because the Doppler effect is stronger than the Lorentz one, and although the Lorentz formula predicts that an approaching end-on ruler is contracted to zero length, this infinite contraction is calculated after the Doppler length-effect. And guess what? The SR "Doppler length" of a ruler approaching at lightspeed (before you apply the Lorentz contraction), is infinite.
note 2:
After applying the Lorentz contraction to this infinite length, it's merely almost infinite <grin>, so again, because the infinite blueshift doesn't happen until the ship speed is infinite using the ship's approaching rulers to measure distance, the SR "lightspeed" is (yet again) equivalent to infinite speed measured with Newtonian distances. This is as it should be, because if Newtonian FTL approach velocities are legal for one observer, they should be legal for both.
Just as in the earlier "redshift" examples, any degree of observed blueshift is always interpreted by the special theory as evidence of motion at less than lightspeed.
One reason why SR is so successful in asserting that objects never travel faster than light is that no matter what amount of shift an approaching or receding object has, whether the observed frequency of radiation is shifted to ~zero or ~infinity, in fact, no matter what data you collect, SR usually has a way of interpreting that data as proof that a velocity is subluminal. There simply isn't any shift value that SR wouldn't map to a velocity less than lightspeed. That makes the idea of a lightspeed limit slightly difficult to disprove
Under SR, there is no lower limit to the amount of time needed to cross a given Newtonian distance. Any object seen to cross a particular region of space in a finite time is automatically assigned a speed less than c, no matter how short the time-value.
Under SR, there is no amount of observed shift in an object that isn't mapped to a velocity less than c.
Under SR, there is no amount of kinetic energy for an object that isn't mapped to a velocity less than c.
Under SR, objects can recede with any velocity without generating evidence that they are receding at >c, because a recession redshift can't make the observed frequency drop below zero.
Under SR, objects can approach with any Newtonian velocity, and their shift is still finite, and their velocity is mapped to less than c.
So although SR places a notional upper limit of c on an object's journey through its environment, in terms of contracted distances, when you re-express this in terms of conventional distances, the upper limit comes out not as c, but as infinity.
Confused yet?
The problem is that SR has been so busy reinterpreting speeds, lengths, distances, times and masses, that there's nothing left for us to look at to see whether the SR "velocity" parameter actually corresponds to anything in real life.
Ironically for a theory that's founded on the principle of pure observation, "relative velocity", the one parameter that's agreed on for all observers under SR, ceases to have any physically-observable meaning, because the hypothetical ruler-contraction effect means that you can no longer use "known distance / measured time" as an agreed definition, even when you cancel out any expected timelag effects. In order to find that original velocity, you are supposed to take the SR redefinitions on trust.
Just think - for decades, people have been reciting that "you can't go faster than lightspeed" argument, thinking that it actually means something deep and meaningful, when it actually translates (in agreed-distance terms), to: "You can't travel faster than infinity".
Kick back your chair, put you feet up, look out of the window and savor the truth of that second statement for a few minutes. It's a beautiful, world, isn't it?
a) You may find it difficult to find a mention of the Doppler ruler-effect outside these pages. It does show up in the occasional mainstream paper. There's a brief mention of it in "Invisibility of the Lorentz contraction" (can't remember the author or date right now). There's another run-through on the relevant DMS page, but using a different Doppler formula and propagation model.
b) b) The Doppler ruler-effect is a sadly neglected topic, not because it's difficult, or unconventional, but simply because most SR discussions are normally limited to the effects of purely transverse motion, where (under the SR model), the Doppler length-changes don't apply. As a result, most physicists seem not to know anything about it.
Here's how it works - suppose that a train of length l is receding from you, and you see the rear of the train to be at position x. The equivalent position of the front of the train is going to be x+l, but the light from it won't have reached you yet. Because the light from the front of the train has an additional time-lag, its viewed position is always slightly more out-of-date than that of the back of the train, and because the train is moving away from you, this extra out-of-dateness (differential timelag) makes the front of the train appear to be closer than x+1, and the train appears contracted as a result. If the train is approaching, then the rear of the train would be showing an older position than the front, and would appear to be further back along the track than you would therwise expect. As a result, the approaching train appears lengthened.
For an extreme example, consider a 1m long ruler approaching the observer, end-on, from a marker point 1 light-year distant, according to special relativity (which uses the Doppler version of the formula). If the ruler is travelling at (illegal) lightspeed, then SR models the ruler as riding its own wavefront. In this case, a point at the front of the ruler appears to the observer to pass from a distant marker-beacon to the observer's position in zero time, because it will arrive at the same moment that it is seen to leave (the events of it passing the startpoint and reaching the finish-point appear to be simultaneous)
However, because the ruler is travelling at finite speed, the front and rear of the ruler don't appear to pass the start-point at the same moment - the rear passes slightly later. As a result, the observer sees the front of the ruler reaching them before the end of the ruler is seen to pass the start-point, and the ruler therefore appears to cover the entire one-lightyear distance. For a ruler approaching at lightspeed, end-on, it's observed length (before you invoke Lorentz contraction) is effectively infinite.
Easy
stuff, but not often taught.
(8): "Probe Chain" theory
(the SR flat-space
model can fail in an inertial system
comprising of more than two objects)
(and not a lot of people know that, either)
If an object accelerated to an "illegal" recession velocity, you wouldn't normally be able to see the c-breaking event happen if you observed the object directly, and yet under SR, the ship's acceleration through a series of intermediate frames means that the ship's recession speed is supposed to be subluminal, and the ship therefore _ought_ to be visible at it's higher (Newtonian) velocity.
Can any events that occur beyond the c-point be observed by a home-based observer or not? How can the ship have two velocities? What is really going on?
The answer comes when we decide how the ship is to be observed. If we choose to view it directly through the vacuum of space, then (under SR), the notionally-FTL portion of the ship's journey ought to be hidden by an event horizon. But if we choose to view the ship through the haze of its own exhaust fumes, then we see something different. The ship now really does appear to be travelling at less than c, in the sense that the event horizon isn't there any more. Light (under simple SR) somehow finds it easier to travel along the expanding exhaust trail than it does across empty space.
Now the signal isn't attempting to leave the ship and cross over to an observer leaving at lightspeed - it's instead travelling through the expanding gas, which produces a near-continuum of intermediate inertial frames between the receding ship and observer.
If the exhaust plume is dense enough for any ray of light to be able to cross the distance by "leapfrogging" between particles that are always receding from each other at less than lightspeed, then the shift incurred by each individual frame transition will always be too weak to bring the frequency quite to zero. Therefore, if the ship is viewed through a series of intermediate frames where the individual frame transitions are all less than c, then the signal will always be able to get through, and the result is what you would have expected if the recession shift had been less than c.
That's the effect that the SR velocity addition formula documents - the existence of intermediate inertial frames removes the original quantified velocity "step" assumed by SR, and replaces it with two or more steps that don't lie on a straight line. By adding the intermediate frames, we are taking a velocity differential and replacing it with a series of consecutive velocity differentials that mark out an approximation of an acceleration curve. A signal sent through these intermediate frames therefore passes through an approximation of an acceleration curve, which, of course, represents a warping of the properties of space under GR.
relative velocity along the path of a light-beam
[missing diagram]
So we can have a completely inertial system, with a number of relay probes travelling at fixed (intermediate) speeds to one another, and the composite signal path isn't flat, but travels along the approximation of a curve, generated by all of the individual nominally-flat frame transitions. Most people don't realize that the SR flat-space postulate can fail in inertial systems comprising of more than two objects, but there it is - add enough straight lines together, and you've got an approximation of a curve. __________ ______ ____ __________ ______ ____ __________________
Let's suppose that the spaceship in the section 2 example accelerates up to and beyond "Newtonian c", so that it would be expected to be hidden by an event horizon. How would the crew ever get a message back home to say that they'd broken the Newtonian lightspeed barrier?
The solution should now be fairly obvious.
The captain of the ship ensures that when the ship sets out, it is stocked-up with a supply of numbered communication satellites, each programmed to receive information received from one of its immediate neighbors, and pass it along to the other, so that the string of satellites forms a chain along which data can be sent, in either direction.
As the ship accelerates, the crew carefully drops the sequence of satellites out of the back of the craft, one by one (being careful not to damage them with the ship's engines), so that the satellites chain marks out a sequence of inertial frames that each have a recession velocity to one another. The key feature of this chain is that although a signal sent along the chain is going to be redshifted, the redshift never reaches infinity, and a signal can therefore be passed along the chain in either direction, even when the two ends of the chain have a Newtonian recession velocity greater than lightspeed. The ship and homebase can therefore still stay in contact, although they would still hear the signals coming from each other to be extremely s.. l.. o.. w.. As the individual probes are separating in space at constant velocity, and feel no acceleration forces, a signal effectively has the same properties as it passes along the chain as if it was passing along a gravitational gradient, causing the probe separation. The remaining recession redshift would therefore appear similar to a gravitational redshift.
Actually, since the shift is "red", the effect is more like a Hubble shift. Anyhow, probe chains seem to give a legal way of sending signals between objects with FTL separation velocities, and therefore also seem to provide a way of sending nominally-FTL signals across any region of space. You just piggyback the light signal through a messenger-object that's going in the right direction, and it gets there faster. I actually thought that the final probe-chain principle would turn out to be a lot more complicated than that, but there you go.
SR Problems: the "Magic Window" problem
(velocity-addition formula, non-flatness of space)
Intro:
SR's claim to validity is based on the assumption that space is uniformly flat and that signal propagation times are wholly unaffected by relative motion. In order to generate the right results from this propagation model, SR uses an additional Lorentz redshift to pull the "propagation" shift figures into line. Under SR, the Lorentz shift is deduced to occur at the affected object, and is not supposed to be a signal-propagation effect.
The "window" problem below shows that an object with a fixed recession velocity can have its SR redshift reduced simply by being observed through an intervening sheet of glass with an intermediate velocity. SR's workaround to this situation is to say that the object's deduced recession velocity is partly dependent on the motion of the intervening glass, according to a special "velocity addition formula" that is used for calculating composite shifts. The new (hypothetical) velocity value is then supposed to be used with the normal SR shift formula.
This workaround (in its current form) is not compatible with the SR propagation model.
Example:
A directly observed object recedes from the observer at 0.8c. It's observed shift is therefore, under SR,
f'/f = root( (1- 0.8) / (1+ 0.8) )
= root( 0.2/1.8 )
= root(0.11111')
Now, this time we are going to look at the object again, but through a thin pane of glass, our "magic window". This "window" is between the observer and the object, and creates an intermediate frame between the other them. The observer and the object are effectively both receding from the window at 0.4c.
We can now combine the two shifts that a light signal picks up travelling from the object to the window, and then from the window to the observer. Each of these redshifts is given by the formula:
f'/f = root( (1- 0.4) / (1+ 0.4) )
= root( 0.6/1.4 )
= root(0.42857...)
Once the signal has picked up two of these shifts and reached the final observer, the total shift on the observed signal is:
f'/f = 0.65465... * 0.65465...
In other words, there is less of a redshift when the signal travels via a third intermediate frame, than when there is a direct path.
The speed and energy of of the signal is somehow "stepped up" by its passage through the intermediate physical frame provided by the "moving" pane of glass. This sounds promising, but such a mechanism is illegal under SR, which assumes that propagation times are unaffected by relative motion.
The amount of Lorentz shift on the observed object is being changed by the altered topology of the experiment. This is difficult to reconcile with the idea of the Lorentz shifts being purely a product of relative velocity, as the path taken by the signal would also be a factor. The observer can still "peek" around the edge of the glass pane and see the original amount of shift in the object, and if a lightbeam is split, with one half sent through the window and one sent directly, both signals would have different Lorentz shifts. This difference would mean that Lorentz shifts are partly path-dependent, so that Lorentz shift couldn't be totally separated from propagation effects.
Motion in the glass reducing both the Lorentz shift and the propagation shift on the object.
By choosing to deliberately reinterpret the deduced recession velocity of the object to a different value when it is indirectly observed, special relativity plumps for option (3).
This workaround reduces both the predicted propagation shift and Lorentz shift of the receding object in the above example. For the "propagation" shift of an object viewed along a straight-line signal path to be reduced when the glass is moving, the motion of the glass must (by definition) be affecting the properties of that signal path, contra SR.
The SR velocity-addition formula is therefore not compatible with the SR propagation model, even though it is used to calculate composite shifts under the theory. For SR to be self-consistent, the velocity-addition formula would have to be replaced with a different formula that preserved the SR propagation shift component, and instead affected only the Lorentz component of an indirectly observed object.
This mistake isn't necessarily fatal to SR, as an alternative correction formula based around "option 2" is probably possible. However, since the inconsistency is fairly obvious, you'd expect an amended correction to already be in place.
The fact that it isn't raises questions as to how accurately SR has been assessed to date. __________ ______ ____ __________ ______ ____ __________________
DMS produces the same class of effect. However, DMS uses interpretation [1], and so, doesn't have a problem with the result. Under DMS, the step-up in energy of the light-signal is caused by the "moving" window "dragging" the signal along in its direction of motion, as an inertial/gravitational distortion effect. A similar mechanism probably exists under GR
The SR velocity-addition law for composite shifts is vtotal = (v1+v2) / (1+ (v1*v2)/c^2). This formula is supposed to provide the user with an "equivalent" velocity of an indirectly-observed object.
Under the SR velocity addition law, the sum of two recession velocities of 0.4c is not 0.8c, but 0.689655...c. This new pseudo-velocity can then be "plugged into" the standard SR shift formula to give exactly the same final (less redshifted) shift result that we calculated above from using the two smaller velocities. That's its purpose. The existence of this formula documents the existence of the "window" effect in SR. It's given on page 39 of Einstein's 'Relativity' book.
While two two arguments above produce the same numerical results when two equal velocities are summed, they don't when the two velocities are unequal and neither velocity is c, so unless I'm mistaken, the standard SR addition formula has some terms missing. Most examples only use equally matched velocities, so the issue doesn't normally come up.
The arguments given on this page can be used as a foundation for "probe chain" arguments. These lead to a reinterpretation of the nature of the SR lightspeed barrier.
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