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Let us imagine you are in a vacuum and after having maintained a speed of 0 km/s (standing still) you accelerate to 297,000 km/s (99%). You know this is now your speed because you have a speedometer telling you so. You then decide to maintain that speed for a while.

With the speed of light is always ~300,000 km/s faster than you, what is preventing you from (again in your reference frame) increasing your speed, as shown by a speedometer, an arbitrary amount faster than ~300,000 km/s? After all, the speed of light will always be always faster.

I feel like length contraction even backs this since it will make your space wheels tinier. You're essentially scaled down and your tiny wheels would have to rotate many more times to go the distance just 1 rotation would have taken you with your non-contracted length. This then would cause the speedometer to relay speeds faster than the speed of light.

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    $\begingroup$ In your (instantaneous) reference frame, your speed is zero. $\endgroup$ – WillO May 16 at 3:00
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    $\begingroup$ How exactly does this speedometer work in a vacuum? I $\endgroup$ – WillO May 16 at 3:03
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    $\begingroup$ Why would spinning your wheels in a vacuum change your velocity relative to anything? $\endgroup$ – WillO May 16 at 3:06
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    $\begingroup$ why can't this be in a car, it's a thought experiment? $\endgroup$ – JEB May 16 at 4:36
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    $\begingroup$ @MichaelKing Relativity is an odd beast. Often intuition falls flat. This can make metaphor tricky. The specific details about how your speed measuring device functions can play a surprisingly large part in resolving confusion at relativistic speeds. Things we assume "just work" in simple settings show their true colors at speed. This is why Einstein was very explicit in what measurements he assumed could be taken (he had a clock and a rod, to measure time and distance respectively) $\endgroup$ – Cort Ammon May 16 at 18:20
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You know this is now your speed because you have a speedometer telling you so.

This is precisely where you hit a (metaphorical) roadblock.

A speedometer must use something outside of your reference frame to measure your speed, as speed inside your frame is either 0 or meaningless (take your pick).

It's measuring the speed of your space wheels.

Your space wheels will never spin faster than the speed of light (where the speed of your wheels is the linear velocity at the edge of the wheel).

It's measuring your current work being put in and converting it to a speed.

Then you are measuring kinetic energy and not speed. There is no upper limit on the kinetic energy an object can have! However when you solve backwards for speed infinite kinetic energy leads to your speed approaching (but never exceeding) c.

$$ KE = mc^2 (\gamma - 1) \\ \gamma = \frac{1}{\sqrt{1 - (v/c)^2}} = \frac{KE}{mc^2} + 1 \\ \sqrt{1 - (v/c)^2} = \frac{1}{\frac{KE}{mc^2} + 1} \\ 1 - (v/c)^2 = \left( \frac{1}{\frac{KE}{mc^2} + 1} \right)^2 \\ v = c \cdot \sqrt{ 1 - \left( \frac{1}{\frac{KE}{mc^2} + 1} \right)^2} \\ $$

If you plug in $\infty$ for KE you should see that you recover $c$.

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    $\begingroup$ RE: "A speedometer must use something outside of your reference frame to measure your speed" This is not strictly true, speed can be inferred from an accelerometer (and there are device/vehicles that do this). I think that the interesting matter here is how does perception (observations of externals), reality (relativity) and inference (acceleration * time) of current velocity differ? $\endgroup$ – RBarryYoung May 16 at 15:05
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    $\begingroup$ @RBarryYoung true - an accelerometer at the surface of the earth would infer you're going faster than the speed of light after a year $\endgroup$ – Señor O May 16 at 15:55
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    $\begingroup$ The problem with inference from accelerometer is that you must make some assumption on how velocities add (and thus integrate). If you use Newtonian rules, you'll get FTL results, but this is not a measurement of speed, so it's not surprising. $\endgroup$ – Ruslan May 17 at 8:56
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    $\begingroup$ An accelerometer lets you infer how much your velocity has changed since you started measuring. It's like watching the money go in and out of somebody's bank account: you can tell how much richer or poorer they were than when you started watching, but without knowing how much money was in there initially, you can't tell their total wealth now. All you can get is wealth (velocity) relative to some reference frame. $\endgroup$ – Geoffrey Brent May 18 at 21:55
  • $\begingroup$ @GeoffreyBrent I think that analogy takes as much effort to understand as just understanding the concept itself $\endgroup$ – Señor O May 19 at 0:53
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In your reference frame you are always at rest, so you can always accelerate to a new reference frame.

Suppose you were at rest in some reference frame. Then suppose you were accelerated to 0.9999999999999999999999999c in that initial reference frame.

In your new reference frame you are stationary. You can now accelerate yourself to 0.9999999999999999999999999c in your new frame.

You are now stationary in a third frame, in which you can also accelerate yourself to 0.99999999999999999999999999c.

And so on endlessly.

However, when you add your series of velocity increases relative to the original frame, you must use the relativistic rules for adding velocities. No mater how many velocities of 0.999999999999999999999999c you add, you will never exceed c.

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Whatever reference frame you are in, and however fast your reference frame moves with respect to mine, the speed limit you observe for objects in your reference frame is always $c$.

For example, suppose you are in the back seat of a car, making paper airplanes and throwing them to the front seat. The fundamental principle of relativity is that you should not be able to distinguish, from your paper-airplane experiment, whether your car is at rest or on the highway moving with constant velocity. However, if you are on the highway, an observer at rest on the side of the highway will still see the paper airplane moving from the back of the car to the front. The roadside observer will measure the paper plane’s speed as faster than the car's, but less than $c$.

In your question you imagine doing this multiple times: there are passengers on the paper airplane, making and throwing paper airplanes of their own. Your instinct is correct: that's also allowed, ad infinitum.

How can this be? Suppose (to be specific) that your car is driving at $u= 30\rm\,m/s$, and you throw the paper airplane forwards at the speed of $v = 3\rm\,m/s$. To find the paper airplane’s speed as measured from the side of the road, you use relativistic velocity addition:

$$ v’ = \frac{ u + v }{ 1 + uv/c^2 } $$

For our specific example, the denominator is

$$ 1 + \frac uc\cdot \frac vc = 1 + 10^{-7}\cdot 10^{-8} $$

so the “Galilean” resultant speed of $\rm 30\,m/s + 3\,m/s = 33\,m/s$ is accurate to about fifteen significant figures. But in your example, the space-car is going at $u = 0.99c$. If you launch your space-paper-airplane at $v =0.1c$, an external observer would measure its speed as

$$ \frac{v’}c = \frac{ 0.99 + 0.10 }{ 1 + 0.099 } = 0.99181 $$

If instead of a space-paper-airplane your projectile were a light ray, with $v=c$, the external observer would also measure $v’=c$. Any other $v<c$ will also give $v’<c$.

Your instinct about length contraction is also valid. If the roadside observer wanted to answer a question about how long it takes for the paper airplane to travel from the back of the car to the front, they would use their observed velocity $v’$ and the contracted length of the car. But that duration result would disagree with the result obtained by someone inside the car, thanks to time dilation.

The non-workability of an absolute speedometer is one of the fundamental philosophical results of relativity, and is worth the time to think about if you haven’t. The original popularization has aged relatively well. A relativistic rolling wheel also presents a puzzle.

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Yes you can reach infinite speed if you call it so:

Let's define as "speedometer" an integrating accelerometer.

If you are in a vacuum and accelerate with 1 g for 1 day, you can call your speed "1 g*day". After all, speed has units of acceleration * time. In SI units, this is 847,584 m/s.

If you continue to accelerate at 1 g for 1 yr, your speedometer will tell you, that your speed is larger than c.

But as others have said, this speedometer is meaningless, as it doesnt tell you the speed with respect to anything beyond your frame of reference.

Even bigger obstacle: There is no theoretical energy source that you could carry that would allow such prolonged acceleration, which means you need an external reference frame to impart momentum on you. Even if you are a light sail and use photons, you will not reach c, because when you look back all light will be infinitely red-shifted and lack energy.

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You do not need a speedometer; as long as you have fuel to "burn" you can add another $\delta v$, and so continue to accelerate. Eventually you will run out of fuel.

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    $\begingroup$ I think the fuel is irrelevant in a thought experiment $\endgroup$ – Polygorial May 16 at 21:34
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    $\begingroup$ You should be able to see through that minor "embellishment"; it was not central to my answer, but i thought it added context. OK, then my point is now that you can accelerate forever, there is no barrier, just keep adding $\delta v$. $\endgroup$ – m4r35n357 May 17 at 8:57
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It is tempting to say that when you accelerate in empty space you will acquire an absolute velocity wrt the universe: you accelerate for a while and obtain an absolute velocity. Even when you readjust the speedometer to include relativistic effects (so that it can never show you speeds higher than the velocity of light), the velocity (speed) shown will be a relative velocity. You could just as well say that the rest of the universe will acquire a velocity relative to you. The difference is that the rest of the universe remains fixed wrt to the cosmic microwave background radiation, while you accelerate in it (the consequence of this is the twin paradox, which says that if you accelerate away from, say, Earth, accelerate backward, and return to Earth, you will find that you have aged less than people on Earth). The velocity you acquire will be a relative velocity though, while the acceleration is absolute. You will never know if it's you who is moving or the rest of the universe. There exists no independent reference frame wrt which you can measure your velocity, as the was once thought: the aether. You might think that the CMBR constitutes this aether, but this is not so (see the answer, by @ArpadSzendrei, to this question).

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These sorts of questions concerning the notion of 'time dilation' eventually demand an explanation in physics or mechanics for the phenomenon originally proposed by Einstein in his seminal paper ''On the Electro-Dynamics of Moving Bodies''[1905] -- in this instance the speedometer is just another variety of clock.  The mathematical argument devised by AE is not difficult to follow after all, provided that, having conceded the postulate of invariant c, the premise is accepted; namely that the light paths ostensible to observers in the moving and stationary frames k and K in AE's scenario actually differ.

Considering then that in all the myriad ways this problem has been posed and addressed since that scenario of a moving light source was first imagined, and its description elaborated mathematically using Lorentz Transformation based on such a postulate of invariant maximal c, no adequate explanation in physics or mechanics has to date been able to account for the effect that, while observers of the moving frame k and its light source situated in the stationary frame K perceive the entire effect of the transit of that light source over its actual distance in K, those moving with that light source in the frame k perceive only that component within that frame k itself, the following seeks briefly to understand this anomaly that these light paths ostensible in those respective frames actually differ.

Clearly then, in order to account for the effect observed in the frame K, that component of light transit constituted by the velocity v of the light source is somehow incorporated into the velocity c of light itself (which one is entitled to suppose is intrinsic to the essence of space), from which moreover it might be concluded that such material motion is in fact an aspect of light, merely a variation on the mechanics of space from which light itself arises as an innate effect; and that this incorporation manifests itself differently for observers in the frames k and K.

The reason that this component v is not apparent to those observers in the frame k is of course that they too are moving with that same velocity, and this is no doubt the original intuition of AE.  What this inevitably suggests however is that these observers, clocks, the light source and the frame k itself constitute a dynamic entity whose capacity to sustain its form and integrity in that motion at v at once impedes the possibility for any objective perception of it with respect to the transit of light itself at c.

What then is the essential sense in which this dynamic entity of an inertial frame in motion, indeed of a singular universal frame of which it is a mere aspect, is to be understood? Here one simply postulates as the context for this imagination the existence of a Unitary Universal Substance [UUS] of which all reality is comprised, including observers in any frame, within which the motion of material bodies, as in this case, is effectively a progressive wave effect -- viz, the progressive aspect of an oscillatory wave dynamic -- deriving in the capacity for that substance continuously to reconstitute itself in a given spatial direction.  In the case of a light source for example, this implies a corresponding dynamic in the motion of electrons.

The postulate of such a UUS need not of itself be troublesome to the receptive intellect.  What is required is only a consistent model of its singular dynamic based on certain corollaries to that proposition itself; namely that in the first place such a substance holds together and coheres with what amounts to a universally 'cohesive force' (moreover with an evident and self-evident proclivity to stability, thus of form, further implying the tendency for cyclical recurrence of phenomena) which, when construed in concert with a singular impetus implicit in such an inviolate universal entity or unity, may be conflated with a property of inertia; a priori this derives in the corollary axiom of a singular universal process or intention implicit in a singular cause.

This 'cohesive force' then may be understood to operate from within the interior spatial depth of such a UUS at every point and importantly in perpetually disparate components whose interaction inclines naturally and inexorably towards a condition of parity or 'cohesive equilibrium': components of cohesive force may be imagined and modeled to operate in axes between such loci of action, therefore distributed in planes* at right angles to such axes at the point of such cohesive interaction, generally corresponding to the geometric relation between a magnetic force and associated E-field. 

All discernible physical forces are only aspects of this fundamental and quite exclusive 'cohesive force', and even 'electric charge' itself is eventually a component of its distribution in planes of cohesive interaction.  *As a guide, imagine the intersection of two expanding spheres at a planar interface.

The property of inertia implicit in the process towards cohesive equilibrium or symmetry, and in the constant irreversible maintenance and enhancement of a condition of universal 'cohesive resonance' or 'cohesive equilibration' between disparate components of cohesive force equivalent to this very tendency, corresponds to 'mass'; so that, for example, the essential relation between that property and the cohesive force itself whose distribution in a given region implies such mass/inertia, and broadly 'energy', is only the function of the rate at which motion within the UUS changes: viz, acceleration.   

It is this cohesive disparity between any and all components of cohesive force which imparts to the UUS its oscillatory or vibrational character, eventually a progressive wave principle as more-or-less regularly incremental resultants in such an effect tend in given spatial directions.  This can be modeled with respect to a condition of the perfect oscillatory regularity of cohesive equilibrium represented by a cubic lattice structure which is therefore appropriate in particular to the analysis of states tending to approach that ideal condition of cohesive symmetry such as H1 gas.

This is the essence of the process of 'reconstitution' of the UUS: that a given configuration of the oscillatory dynamic of cohesive relations will tend naturally to recur with an incrementally progressive aspect which depends on resultants in cohesive disparity -- viz, 'cohesive polarities' which are components of cohesive force --, the limits of the regularity in which define the wavelengths of visible light (essentially an 'interference effect' between fundamental vectors of cohesive polarity in such a model).

Obviously then, nothing may move relative to a unitary universal substance; all motion and relative motion is of and within that substance itself.  This motion, indeed all motion, is inherently wave motion -- which is to say that all motion within the UUS arises as an oscillatory effect between points of disparate cohesive potential in transient existence as disparate resultants of prior cohesive interaction within the UUS; such that the more-or-less regularly incremental progression in such an effect constitutes wave motion and corresponding velocity in a given direction, maximally c in the limiting case in which this incremental progression in a fundamental interference relation between two disparate archetypal components of cohesive force corresponds to a specific wavelength of visible light in vacuo.

It is for this reason then, in the cohesive mechanics of such a UUS and its unitary wave (interference) dynamic or principle that the resultant motion of electrons constituting a light source moving in a given direction cannot be apprehended by observers moving in the same way with the same relative resultant; while to those observers stationary in the frame K relative to that source, the entire effect is perceptible as an integrated resultant within the limiting context of a maximal resultant c.  The velocity v of the frame k and the light source becomes an aspect of c, the wave velocity intrinsic to the UUS, as if overlaid upon it; while the velocity c of light is itself eventually only the manifestation of the optimising tendency of the cohesive dynamic to attain to an ideal condition of cohesive equilibrium or symmetry between any pair of two mutually cohesive loci within the UUS, a metaphysical impossibility

It is important to recognise here that since we too are composed entirely of such a substance, its ultimate nature can never be objectively discerned; in a general sense its optimal condition of homogeneity is appreciable as 'empty space' (with its VEV etc.).  At the same time however, since the essential principle inhering in such a substance is one of 'cohesive resonance' between its constituent components, effectively components of 'cohesive force', its dynamics are readily perceptible at every turn, and similarly amenable to understanding.  Indeed, since the perception of the operation of the fundamental cohesive force through mechanisms of 'perceptual resonance' -- viz. perceptual consciousness -- represents the sole means of apprehension of this UUS, then that substance or fabric may be regarded as effectively equivalent to that ubiquitous force.  This intuition was also expressed by Faraday.

Clearly though, nothing in the preceding argument permits that the regularly oscillatory mechanism of clocks of any kind actually proceeds more slowly when in motion; and nothing of the sort is to be inferred from AE's original scenario.  This idea has somehow gathered conventional and popular credence for want of the capacity for the orthodox perspective of physics as it stands -- that is, a body of convention which refuses to countenance the notion of a UUS requisite to any comprehensive theory of reality -- to explain the effect that observers of a light source from diverse inertial frames in relative motion differ (in the manner described by Lorentz transformation of coordinates between them).

The reasoning here constrains itself to explaining the mechanics of that phenomenon according to a consistent model of the cohesive dynamics of a UUS incorporating all such frames.  That model is a dynamic purely geometric model based on the ideal of 'cohesive equilibrium' towards which the unitary 'cohesive dynamic' within that substance inherently inclines -- according to a principle of increasing 'cohesive resonance' between constituent components --, to which an appropriate time scale, essentially a fundamental oscillatory period which must be argued to inhere within such a universal unity, is then ascribed.

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What actually matters here is not length contraction, but time dilation. Assume you want to travel between Earth & Alpha Centauri. Handwave away the problems with fuel & reaction mass, and assume you can use sufficiently advanced magic to accelerate at a constant 1 g. Then your speed is simply the distance divided by the length of time it takes you to make the trip.

But your time aboard the ship is not going to be the same as the time experienced by observers back on Earth. You will have (per Google) spent about 2.3 years on the journey, and thus travelled faster than light for part of it. However, if on arriving you set up your powerful laser and beam a "We made it!" message back to Earth, they will get it about 8.9 years after you left (again, per Google), meaning that to them you never went faster than light.

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    $\begingroup$ If constant, indefinite 1g acceleration were possible, then you could do FTL signalling to Alpha Centauri. Your answer implies that c is only a limit for light, but that you could instead have a message rocket made of the same unobtainium that sends the response in 4.6 years total elapsed time. $\endgroup$ – Lawnmower Man May 16 at 19:46
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    $\begingroup$ The "proper time" for a one-way trip to Alpha Centauri, in which the rest-frame acceleration is always $1g$, may indeed be less than four years. But no observer believes this involves faster-than-light travel. The traveling observer sees the travel distance reduced by length contraction; the stationary observer disagrees with the travelers' clocks, thanks to time dilation. $\endgroup$ – rob May 16 at 20:30
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    $\begingroup$ @rob: That is entirely a matter of perspective. If I know that Alpha Centauri is 4 light years away, yet I cross the distance in 2.3 years of my personal time, then FROM MY POINT OF VIEW I must have have travelled at more than an average 3*10^8 m/s. Now if I happen to be familiar with relativity, I can use the theory to explain why this happens (length contraction/time dilation), but if I happen to be travelling in a spaceship without windows, I will not see any evidence of this. $\endgroup$ – jamesqf May 17 at 2:18
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    $\begingroup$ @Lawnmower Man: That is simply wrong. You can travel to Alpha Centauri experiencing about 2.3 years of time, while folks back on Earth experience 4 years. Then if you immediately turn around and head back (or send the message rocket), you will experience another 2.3 years, while Earthlings experience 4 years. So Earth folks see the round trip as taking 8 years, you see it as taking 4.6 years. This is the well-known "twin paradox": en.wikipedia.org/wiki/Twin_paradox $\endgroup$ – jamesqf May 17 at 2:18
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    $\begingroup$ You write “Alpha Centauri is four light-years from Earth” as if that were not a matter of perspective. But that’s only the distance in Earth’s rest frame, which is no more privileged than any other frame. From the traveler’s point of view, the traveler is at rest and the rest of the universe is moving. You are making a similar external-speedometer error as the asker of the question. It is a subtle error and you’re in good company; it usually takes an hour for me to help a prepared student get to the other side of it. $\endgroup$ – rob May 17 at 11:00
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For all practical purpose, speed is the distance you travel divided by the time it takes. What is often understated is that a traveler can indeed, from its own perspective, travel faster than light.

For example, travelling fast enough, someone could reach Alpha Centauri (4 light years away), with only 2 years passing on his clock. He can then correctly conclude that he traveled at twice the speed of light.

Of course, for earthlings, more time will pass, and from their point of view the traveler will never travel faster than light. This is often incorrectly used to explain why we would need generation ships, but for the people onboard, the speed of light "limit" doesn't really apply, what really limits their effective speed is propulsion.

EDIT:

This definition of "speed" here is just $v = \frac{\Delta x}{\Delta t}$, both $\Delta x$ and $\Delta t$ are measured at rest (i.e. sitting on Alpha Centaurii).

My point is just that you can indeed keep accelerating to make your trip shorter (for you). Most people thinks it is not possible, and IMHO it comes from the fact that it is almost never clearly stated in our explanations.

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    $\begingroup$ very wrong. In your answer you are measuring speed by dividing distance traveled in the rest frame over time passed in the moving frame. If you did it consistently, calculating it either in the rest frame, or in the moving frame, you'd find an answer less than $c$; the inconstistent mixing of reference frames didn't take into account the lenght contraction experienced by the moving observer. $\endgroup$ – tbt May 18 at 9:14
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    $\begingroup$ That is why I define what I mean by "speed" first: at the end of his travel, the traveler is 4 light years away from his starting point, and 2 years have passed for him. $\endgroup$ – agemO May 18 at 9:34
  • $\begingroup$ "In your answer you are measuring speed by dividing distance traveled in the rest frame over time passed in the moving frame" The traveler is at rest when he is at his destination, his clock displays "2 years", and if he decides to measure the distance he just traveled using a reflected light beam for example, he will measure 4 light years. $\endgroup$ – agemO May 18 at 9:45
  • $\begingroup$ By the way there is no such things as "time passed in the moving frame", just clocks. $\endgroup$ – agemO May 18 at 11:48

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