When it is said that no object can exceed the speed of light $c$ in vacuum, I have some misunderstanding about this statement. Does exceeding the speed of light mean exceeding the speed of light relative to all other objects in the Universe?
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$\begingroup$ Good answers here: physics.stackexchange.com/q/485999/246973 $\endgroup$– WookieCommented Dec 5 at 13:16
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4$\begingroup$ Yes. No object can exceed the speed of light in vacuum relative to any other object. $\endgroup$– SteevenCommented Dec 5 at 15:24
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1$\begingroup$ If you measure with the proper units (i.e. proper time, $\tau$, instead of regular coordinate time, $t$), the speed of light is infinite (in the sense that $\lim_{x\to c}x=\infty$). You can't exceed infinity. $\endgroup$– controlgroupCommented Dec 6 at 3:13
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$\begingroup$ @Steeven I can see Sirius at night, and I know it is more than 8 light years away. If I stay still and watch it for long enough, I can see that it has moved across the sky, equivalent to a full circle in about 24 hours. Even allowing for its declination, Sirius appears to me to be moving across the sky at several thousand times the speed of light. Obviously this is due to the rotation of the Earth, but it is an argument for careful language. $\endgroup$– HenryCommented Dec 8 at 2:18
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1$\begingroup$ The speed of light is not relative to any other object. It's absolute. That's why everyone, regardless of their movement, measures the speed of light as a constant. But since we can move it must be that time and space can warp. This is how the Euclidian definition of space died. $\endgroup$– candied_orangeCommented Dec 8 at 13:33
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6 Answers
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It means the following: suppose the object has a good clock and a light that can flash in all directions (and for simplicity assume everything is transparent to this light). At $t=t_0$ on the object’s clock it flashes the light. Then at any $t>t_0$ the points where the light from the flash are located form a closed surface. The object will always be on the inside of that surface, regardless of how strongly it accelerates or how else it moves.
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3$\begingroup$ To be clear to the OP, the answer to the question in the post is "yes". $\endgroup$ Commented Dec 5 at 13:46
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$\begingroup$ @BioPhysicist. Answer is NO. It means exceeding $c$ relative to any single entity in the universe, since postulate of constancy of $c$ guarantees that there are none of those. $\endgroup$ Commented Dec 5 at 13:48
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2$\begingroup$ This is why I deliberately did not give a yes or no answer $\endgroup$– DaleCommented Dec 5 at 17:43
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1$\begingroup$ @AgniusVasiliauskas I guess we are splitting hairs here, since I'm not picking up on the difference. Dale, yes that's probably best; I was debating on saying anything since I feel like there are multiple ways to interpret the question, but I convinced myself into one reading I guess $\endgroup$ Commented Dec 5 at 19:48
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1$\begingroup$ @BioPhysicist Difference is that exceeding speed of light wrt one object does not mean exceeding $c$ wrt to all objects. Consider co-moving pair of objects which both exceeds light speed relative to some background, but between themselves they do not exceed $c$, because they are stationary wrt to each other. So OP mentioned question has broader assumption which is not necessary to test wreckage of speed limit $c$. $\endgroup$ Commented Dec 5 at 22:32
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Fun fact: Suppose that astronaut Allen has a laser mounted to his spacecraft that sends pulses of light off toward infinity. Allen performs an experiment proving that the pulses travel at speed, C. Meanwhile, astronaut Beth zooms past Allen, chasing his laser pulses, from Allen's perspective, at half the speed of light (C/2). She performs the same experiment, and she finds that Allen's pulses are travelling, relative to her own spaceship, at the same speed, C.
This "fun fact" was predicted by James Clerk Maxwell in 1865, and it shook up the physics community because if it's true (and as far as anybody knows today, it is true) then it has some weird consequences. More on that in a moment,† but first...
...If Beth doesn't blink, she might see Charles zoom past her at half the speed of light (her perspective). If Charles performs the same experiment, he too will see that, relative to his ship, Allen's pulses are travelling at C. Same goes for Deirdre and Evan and Frieda and George and Heather, etc. Each of them is going 1/2 the speed of light faster than the previous one, and yet all of them measure the pulses as having the same speed.
From Allen's point of view, that means that none of them was able to outrun the laser pulses. As far as he can see, no matter how fast each astronaut is flying relative to the others, none of them can fly faster than a beam of light.
†The whole point of Einstein's special theory of relativity was to explore and catalog all of the weird consequences of Maxwell's prediction. Einstein's work basically tied together and patched up previous attempts by other physicists, including, especially, Hendrik Lorentz.
Lorentz contributed the idea of a transformation of spacetime coordinates that allows us to see what relative motion looks like from different perspectives. The Lorentz Transformation is designed such that light-like world lines are eigenvectors of the transformation, which is a fancy way of saying that the transformation never changes the speed of light.
Most everything else that is predicted by the special theory of relativity—time dilation, length contraction, effects on mass and energy—is tied in one way or another to the Lorentz transform.
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10$\begingroup$ Nitpick on the history: While Maxwell's equations were first written down in 1865, they can equally well be viewed as describing waves in a fluid ("luminferous aether") whose speed is measured relative to the fluid's rest frame; this was Maxwell's (and everyone else's) interpretation for some time thereafter. It wasn't until later that Einstein took seriously the idea that Maxwell's equations are frame-independent. See my answer here for further details. $\endgroup$ Commented Dec 5 at 14:05
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1$\begingroup$ "Meanwhile, astronaut Beth zooms past Allen, chasing his laser pulses at half the speed of light (C/2)." To be clear, she is traveling at half the speed of light from Allen's perspective, right? Because all motion is relative? $\endgroup$– MJ713Commented Dec 6 at 1:41
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1$\begingroup$ @MJ713, Yes. Thank You. Answer updated. $\endgroup$ Commented Dec 6 at 2:41
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Does exceeding the speed of light mean exceeding the speed of light relative to all other objects in the Universe?
I think a better statement is exceeding the speed of light means exceeding the speed of light relative to any other subluminal object in the universe.
An observer co-moving with any object in the universe will observe the local speed of light to be c. Anything exceeding the speed of light would be exceeding the speed of light according to any possible subluminal observer.
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According to the postulate of special relativity (and to the many confirming experiments) light speed $c$ does not depend on the relative speed of a moving source or the detector. Hence, no matter how you or light source is moving - you always will catch light going towards you at the same speed $c$ (if in vacuum for sure, because no medium condition is also assumed, as per Maxwell equation $c=1\bigg/ \sqrt{\varepsilon _{0}\mu _{0}}$).
Consequently, your speed measured relative to any inertial reference frame (any entity in the universe) will be less than $c$, since postulate of $c$ constancy guarantees that there are none of such objects. As all inertial reference frames are equal in the sense of Physical laws invariancy between them,- this speed limitation applies to ANY object speed measured relative to any other inertial reference frame in the universe.
NOTE:
In reality if you will measure some very distant galaxy $\mathcal X$ receding speed from us, you may get it's speed as $v_{r} \gt c$. BUT it doesn't disprove mentioned special relativity postulate, nor special relativity itself since this proves that space between us and that galaxy expands faster than $c$ (proven by Hubble law $v=H_0D~$: the further away galaxy is- the faster space expands between that galaxy and it's light detector. When galaxies distances are $\gt c/H_0\approx 14 ~Gly$,- space expands faster than $c$ between them and us).
This is not forbidden by special relativity, because it does not relate to a galaxy physical movement through space , but space expansion speed between galaxies itself.
answered Dec 5 at 14:22
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$\begingroup$ Imagine: two galaxies (or any observable object for the matter) going at (near) light speed away from us, in a straight line, but opposite direction. We'd observe them as both moving away at (near) light speed. But at what speed would they see eachother moving away? intuitively I'd say they're moving at (near) 2 times light speed away from each other. How does that work? $\endgroup$– ExcellorCommented Dec 6 at 10:22
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3$\begingroup$ @Excellor How to translate between different frames of reference is exactly what relativity tells you; it turns out that translation isn't just the simple vector addition we're used to from newtonian mechanics, so you can't just intuively say "near c + near c = nearly twice c". The calculations come out so that each of those objects moving near light speed away from you would measure the other moving away faster than you measured it, but still below light speed. $\endgroup$– BenCommented Dec 6 at 13:03
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1$\begingroup$ @Excellor In special relativity relative speeds of objects are added differently than in Newtonian physics, namely as per formula $$u={v+u' \over 1+(vu'/c^{2})}.$$, (instead as of $u=v+u'$) BUT, if you speak about distance increase between galaxies due to space expansion (Hubble law mentioned in the post),- this is not covered by relative speed addition formula at all, because galaxies in reality does not move in space, but space expands between them at a rate $V$ which when "drags" galaxies with it together to the opposing sides (imagine space expansion like some some of "drag force"). $\endgroup$ Commented Dec 6 at 15:23
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1$\begingroup$ @Excellor Good example of space expansion would be,- draw a couple of dots on the un-inflated balloon and start blowing it,- you will see that as balloon expands,- these dots have some rate of getting further and further apart, but as you know- physically these dots does not move across balloon surface, just surface expands itself. Similarly, imagine a space being 3D hypersurface of some 4D object, and apply inflating balloon analogy to understand that in reality galaxies are fixed points in space, just "space grid" increases resolution at each time point. $\endgroup$ Commented Dec 6 at 15:48
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Causality ensures that cause precedes effect, so any event can't influence others outside its future light cone. The speed of light is the universal speed limit, as dictated by special relativity, because it ensures spacetime intervals remain invariant for all observers. If communication were faster than light, it would violate causality, allowing effects to precede causes in some reference frames. This would lead to paradoxes, like sending information to the past or a glass breaking before it has fallen.
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Consider two objects with mass: A baseball and a train.
A train has more mass than the baseball. So it is harder to move the train than the baseball. It requires more power to move the train.
When an object moves faster it actually increases in energy and energy and mass are closely related. The object becomes harder to move when it is already moving fast it becomes harder to move the object even faster. The inertial mass of the object increases when it moves.
If you are familiar with limits in math, the math happens so that an object with mass cannot ever reach the speed of light, because that is sort of like the "limit" it diverges to that limit.
It becomes increasingly harder to speed up an object that is already moving fast, and this pattern continues and the power required to speed up the object keeps increasing as the object's speed increases and it becomes impossible to increase it anymore before it reaches the speed of light. Although, light itself has no mass.
The speed of light in a vacuum is specified because light can hit things and bounce off of stuff, that's why the sky is blue because violet/blue light bumps into the sky more often than other light. And bumping into stuff can appear to slow light down, because there were bumps along the road, but the actual speed of light does not change, the distance of the path is simply increased and it's not a straight line. UV (ultra violet) light bumps into ozone even more than regular violet light which stops it from entering the world too much but because ozone is depleted due to climate change more UV is making it through these days.
The speed at which you move also decreases the frequency at which your clock ticks, called time dilation. This is another "limit" case scenario, as your speed approaches the speed of light, your rate of time approaches 0, so you would not be moving in time if you went the speed of light and you'd go backwards in time if you went faster than light, but you can't. It's a limit scenario. Your actual speed relative to you can simply not reach the speed of light, that is what people mean.
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2$\begingroup$ "The inertial mass of the object increases when it moves" No, its momentum increases, and Newton's second law $F = dp/dt$ then says that a larger force is needed to change this momentum, be it in direction, magnitude, or both. Either way, your answer is very unfocused, and I am struggling to see how it answers the question at all. $\endgroup$– paulinaCommented Dec 5 at 18:34