Why Is There a Cosmic Speed Limit?
Why Is There a Cosmic Speed Limit? |
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Written in November 2022
The “Cosmic Speed Limit” is responsible for what we know as “the speed of light”. Some people with an interest in both physics and cosmology wonder why there should be an ultimate limit on velocities in the universe. This page answers why there is such a thing as a “Cosmic Speed Limit”, resulting in what is known as “the speed of light”. Unlike other articles of mine, in which I proceed in a roundabout way before I home in on the main subject, in this one I’ll get straight to the point by immediately answering the titular question. After that, I’ll offer further and further layers of explanations. I’ll follow this approach because I assume that the reader who ponders this question already possesses some necessary background. (Without any background, the question does not even occur to a person’s mind.) So, without further ado, I will now offer the short answer: There is a speed limit in the universe (“Cosmic Speed Limit”, henceforth CSL) because whenever the dimension of time is considered in conjunction with any (or all) of the other three macro-dimensions (which comprise what we call “3D-space”) the geometry of the resulting space-time is hyperbolic, not Euclidean. What does that even mean?
Our universe has (as far as we know) four macro-dimensions: the three that form what we know as “space”; call them s1, s2, and s3; and one that we perceive as “time”; call it t. On a large scale, any proper subspace of our 4-D spacetime that includes s1, s2, and/or s3 but not t has the Euclidean metric; but any subspace that includes t and any of s1, s2, and/or s3 has the hyperbolic metric.
In a space with a hyperbolic metric, a
hyperbola
(red curve) is the locus of the points that are at equal distances
from the origin. For those readers who have little background in mathematics and wonder what a metric is, it’s really something very simple: it’s a way to measure distances in (a) space. We’re all familiar with the Euclidean metric, which employs the formula of the Pythagorean theorem to give us the distance d between points (x1, y1) and (x2, y2):
The above formula applies on a 2-D space with dimensions X and Y. If the space is 3-D with dimensions X, Y, and Z, the formula for the distance d between points (x1, y1, z1) and (x2, y2, z2) is correspondingly simple:
(To true mathematicians: please don’t write to me telling me that things can be more complex. I know about metric spaces. I know how a metric can be more abstract than the above simple examples and what properties it must satisfy in order to be called a metric. Here I’m trying to explain things in a simple enough way to non-mathematicians.) And here is how a Euclidean space could be depicted:
In a space with a Euclidean metric, a
circle
(red curve) is the locus of points that are at equal distances
from the origin. So: we take a measuring tape or ruler and measure distances in our 3-D space the way we do because our 3-D space is equipped with the Euclidean metric (at least locally, and away from strong gravitational fields), and the Pythagorean Theorem applies. But we can have a space that’s hyperbolic; i.e., one in which the formula for measuring distances is not exactly the Pythagorean one, but this slight modification of it (in a 2-D space with dimensions s and t):
Notice the minus sign in front of dimension t! That minus sign turns the space of dimensions s and t into a hyperbolic one. And if we have a 4-D space with dimensions x, y, z, and t, then again we have a hyperbolic space if the formula for the distance d is:
That’s the hyperbolic formula for the distance between points (x1, y1, z1, t1) and (x2, y2, z2, t2). Notice how, again, the minus sign in front of dimension t turns the space into a hyperbolic one with respect to that dimension. Actually, as we shall see in the next section, it’s not crucial that the dimension for t has a minus sign. What’s crucial is that the three dimensions for space and the one dimension for time have opposite signs. In other words, the above equation could be written as follows:
And that’s the equation we’ll derive in the following section. What does all this have to do with the CSL, the Cosmic Speed Limit? It directly explains why there is a CSL. In a hyperbolic space there are those two asymptotes, the two diagonal lines that form an X. Roughly speaking, those two lines “are” (or determine) the CSL. But why? Because every entity that moves at uniform speed along a straight path (or even one that remains stationary with respect to one of the dimensions) corresponds to a straight line in space (whether Euclidean or hyperbolic). And when the space is hyperbolic, the speed of the entity corresponds to the slope of the straight line. And the limiting slope in a hyperbolic space is the slope of the asymptote (any of the two; to be specific, this one / we’ll call it the “primary” asymptote). The slope of the asymptote in geometry corresponds to the CSL in physics, i.e., to the speed of light. Note, in addition, the following:
It is that simple. Notice that there are no such “special lines” — no asymptotes — in a Euclidean space. Consequently, there is nothing like a CSL in Euclidean spaces.
A space with the hyperbolic metric is what we learn to call a “Minkowskian space” in relativity. But, “Hold it!” a physicist might exclaim. “The metric of our universe is not Minkowskian! It is Minkowskian only in the absence of matter, where special relativity applies! But if matter is present (as in our real universe), then we have Friedman’s metric, and general relativity applies! Therefore, what you claim holds in an unreal universe, devoid of matter.” Yes, true. But to explain the existence of the CSL, the Minkowskian space of special relativity suffices. Allow me please to explain this. Consider the following analogy: suppose you’re told that the shape of the Earth is like that of a sphere. And then you exclaim: “Hold it! The shape of the Earth is not a sphere! The Earth has all those mountain ranges, valleys, crevices, gorges, craters, ocean basins and what-have-you! How can you call that a sphere?” Yes, true. But the mountain ranges etc. distort the shape of the sphere at a very local level. At a global level, our planet is still spherical to a great approximation. (Ignore its spheroid, pear-like shape for the sake of this analogy.) Likewise, in our universe, matter (as where massive objects such as planets and stars exist) distorts the Minkowskian nature of space-time at a very local level. In intergalactic space, where practically no matter exists, the speed of light (which equals the CSL) is still a limiting speed, and it requires an explanation. Actually, we don’t need to go as far as intergalactic space for the absence of matter. Even within our solar system, or between the Earth and the Moon, space-time is bent so negligibly by the masses of nearby objects that it is Minkowskian to a very great approximation. Approximating the space of general relativity locally with a Minkowskian space is akin to approximating any curved smooth surface locally with a Euclidean plane; and, as every mathematician will assure you, there’s a theorem of Calculus in which it is proved that such an approximation is always possible. (The approximation is the tangent plane to the surface at the surface point where the approximation is locally made.) Now, why is our space-time globally Minkowskian, i.e., hyperbolic in time (its local dents due to massive objects notwithstanding)? That’s an even deeper question, which I plan to answer in another text. (It has to do with human cognition and the anthropic principle.) But in this one we’re dealing with the question: given that our space-time is globally Minkowskian, why is there a CSL (hence, a speed-of-light limit)? This question has already been answered with what I wrote above, but for the sake of clarifying and embellishing the answer with some details for the interested reader, allow me please to go on.
Let us now examine the proposition: “Every object that moves at uniform speed along a straight line corresponds to a straight line in a space with the hyperbolic metric, and the speed of the object corresponds to the slope of the straight line.” Suppose we consider the slope of the straight line that we called an “asymptote” earlier. The property that characterizes the asymptote is that every piece of it corresponds to an object that moves in a way so that the space travelled by it is equal to the time it took for it to travel, assuming that the units along the t and s axes are suitable for the line to be called an “asymptote”. This is shown in the diagram that follows. (The drawing of the hyperbola has been omitted to make the diagram less cluttered.)
Here, the red piece of straight line, which is part of the primary
asymptote, corresponds to an object that has travelled an interval
of space equal to s2 – s1 while
it took time t2 – t1 for it to
travel that interval. Because this is the asymptote, s2
– s1 = t2
– t1.
Or, if we use s for s2
– s1
and t for t2 – t1, we may
write: s = t. This can happen only along the
asymptote. Now, paying close attention to the
intervals s
If the unit of time is the second (s) and the unit of length is the meter (m) then in our universe c is equal to 299,792,458 m/s; in other words, nearly 300,000 km/s. If, however, we used the year as unit of time and the light-year as unit of length, then c would be equal to 1 (light-years per year). Thus, the value of c in certain units is a matter of human convention. But the human-independent fact is that there is a “special” slope for which equation (1) holds, no matter what we decide that the arithmetic value of c should be in our arbitrary units. That slope is the slope of the asymptote. In conclusion: c, which is the speed of light, has the strange value 299,792,458 m/s (that looks not just strange but also arbitrary) because our units of measurement are arbitrary. The universe is not interested in whether we measure time in seconds and lengths in meters, feet, yards, miles, furlongs, or stadia. The universe “says”: “There are these asymptotes (because spacetime is hyperbolic, not Euclidean). Whatever moves along those asymptotes moves with what you humans call the speed of light. Measure that in whatever units you wish; its nature won’t change, because it corresponds to the fixed slope of the asymptotes of the hyperbolic spacetime; that is, of the 4-D continuum that forms the canvas of my own existence.”
In geometry, (t1, s1)
and (t2, s2)
are called “points” (of the 2-D space). In physics, they
are called “events” (of a 2-D universe, but if s includes all
three spatial dimensions then we talk about
events of our 4-D universe). In physics, events
that satisfy equation (1) have a “lightlike” relation with respect
to each other; meaning that a photon of light starting off from s Equation (1) is very close to giving us the
distance between two events. We simply
need to manipulate it a bit to arrive at a proper equation for the
distance. First, we square both of its
sides: And then we bring both operands to the same side: Equation (1´´) says that the distance between two
lightlike events (t
Solving for d in equation (1´´´) we finally get the equation that gives the distance of the two events:
Equation (2) is essentially the same as the one we saw in the introductory section. Only its signs are reversed with respect to t and s. As we’ll see in a minute, we did this in order to have a positive quantity under the square root for the cases of objects that, like us humans and our world of massive objects, move at speeds less than the CSL (the speed of light). Let’s see two events (t1, s1) and (t2, s2) for which | s1 – s2 | < | t1 – t2 |. In other words, those two events correspond to the starting and ending point of an object that moves at uniform speed slower than the CSL, because the slope of the line that connects the two events is smaller than the slope of the asymptote:
The red lines drawn in these diagrams are called “world lines”. The red line in the above diagram represents an object starting at event (t1, s1), moving at uniform speed, and ending at event (t2, s2). Since | s1 – s2 | < | t1 – t2 |, this object moves slower than the speed of light (or: the CSL). And this, by the way, is the explanation for the choice of showing the t axis horizontally and the s axis vertically, rather than the other way around: because in this way it is conceptually easier to think of slopes smaller than the slope of the asymptote corresponding to objects moving slower than light. Conversely, slopes larger than that of the asymptote are for entities moving faster than light.
The same would be true if the red line was flipped vertically, causing s1 and s2 to exchange their positions on the s axis; still it would be true that | s1 – s2 | < | t1 – t2 |. The only change would be that the object would be moving from s2 to s1, i.e., in the opposite direction. Objects that move slower than the CSL are called “timelike”. And because obviously this holds also for humans, it follows that we, too, are timelike objects, together with everything that surrounds us. We have an extremely longish existence in time, whereas our existence in space is very-very limited — an “almost nothing”. Repeat: we are timelike entities, even if we’re unaware of that fact. Now let’s see the line of something that moves faster than light (always along a straight line and at uniform speed):
In the above diagram we see that | s1 – s2 | > | t1 – t2 |. Therefore, since the entity extends more in space than it does in time, it moves faster than light.
Entities that move faster than the CSL are called “spacelike”. The red line in the previous diagram shows a spacelike entity.
If an object moves at a non-uniform speed, its line will be a curve; but still, the slope of the tangent line at each point of its curve will be less than that of the asymptote. Only light moves at the CSL (in vacuo — this needs not be mentioned henceforth) and is lightlike. Or, in terms of our diagrams, only light has the slope of the asymptote. Why a “Limit”?But why is the CSL a limit of speeds? If there are entities with slopes smaller, equal, or larger than the slope of the asymptote, the CSL could be called a “special case”, referring to those particular entities that have slopes exactly equal to that of the asymptote. The reason is that all entities that already have slopes smaller than that of the asymptote (the timelike ones, those that travel slower than the speed of light) cannot increase their speeds (“accelerate” is the technical term) so as to reach exactly the slope of the asymptote, because to do so would require infinite energy. For them, the CSL is indeed a “speed limit”. And, unlike our familiar speed limits that the traffic police sets up on roads, breaking the CSL would require breaking the laws of physics — an impossibility. Since “timelike” refers to us and our surroundings, this is an absolute speed limit for us, humans, and for any technological item we may manufacture. Likewise, all spacelike entities (assuming any exist), which already travel faster than light cannot decrease their speeds (“decelerate”) so as to reach exactly the CSL because to do so would again require infinite energy. The energy of a spacelike entity decreases when its speed increases, tending to zero as its speed tends to infinity. Why does this happen? Why is “infinite energy” required for a moving entity to reach exactly the CSL, either accelerating (“from below”) or decelerating (“from above”)? Here is the formula that relates energy E and speed v, in relativistic physics:
We see that, on one hand, if v < c (timelike entities) then as v → c “from below”, the square root at the denominator → 0, therefore we get E → ∞. On the other hand, if v > c (spacelike entities) then the square root at the denominator takes on imaginary values; and if we assume that v → c “from above”, the denominator → 0i. In that case, if we want E to be a real number, mathematically thinking, m (the mass of the faster-than-light entity) must also be imaginary — although that is a matter of debate among physicists. In any case, once again, E → ∞. The only entities that already have the CSL and are observable by us are the all-too-familiar to us photons, the “light particles” (more formally: quanta of the electromagnetic field), which have zero mass and thus travel at the speed of light. Every entity that has zero mass is obliged (by the laws of physics) to be moving at the CSL. Because light (photons) was the first entity that was observed to have the CSL, the CSL was called “the speed of light”. But there are other entities that have zero mass and thus travel at the CSL, such as the gluons, which bind particles in atomic nuclei and are quanta of the strong nuclear force. (For a list of massless particles that exist, or are thought to exist but have not yet been detected, you may check here.) That’s why the terms “CSL” and “speed of light” have been used interchangeably in this text. The point is, once something is at the CSL, it cannot change its speed to a different value. And once something is not at the CSL, it cannot change its speed to reach it. Hence, the CSL is the Cosmic Speed Limit. Absolute Spacetime Described by Special Relativity??The previous diagrams and the preceding discussion, where I claimed “The above diagram is the real hyperbolic space” (in Note 2) seem to imply that they depict a universe in which special relativity reigns supreme. That cannot be our universe! In special relativity motion is relative, there is nothing like a preferred frame of reference for all observers. Whatever happened to general relativity? If the way I present things above is correct, what was the point of Einstein proposing his grandest theory ever, the theory of general relativity, in 1915, as a sort of “correction” of special relativity? General relativity is fine and well, and applies in our universe, at least on a macro-scale (its incompatibility with current quantum theory notwithstanding), but it is useful where gravity is present. Where gravity is negligible, where matter is so scarce that it can be practically ignored, and hence, where the fundamental formula E = mc2 is not too useful (on a macro-scale), general relativity is something of an overkill. And such is the situation when we look at the universe as a whole. Yes, massive objects such as stars are very relevant and important for us humans, and we want to know how the quartet space-time-matter-energy behaves in their vicinity; but when we look at the universe as a whole, huge masses are but a mere footnote! Even where gravity is very relevant for us, as it certainly is on the surface of our planet, just how much slower do our clocks run due to the gravitational field of the Earth with respect to clocks in the outer space? Answer: by an amount so negligible that we could safely ignore it, had we not had the need to obtain extremely high precision in our GPS systems over long periods of time. Who, if the person is not an astronomer, really cares about whether the light rays of a star are ever so slightly bent by the gravitational field of the Sun (which can only be observed during a total eclipse, to boot), or about the gravitational lens phenomenon? (I am talking about macro-world observations involving gravity; relativity is useful to — and even necessary for — experimental quantum physicists, but special relativity suffices for all practical purposes in the experimental quantum world.) In the macro-world at the scale of our entire observable universe, super-massive stars, neutron stars, and black holes are akin to craters and mountain peaks on our planet — an analogy I gave earlier, in the introductory section. To harp a bit more on this analogy, suppose you’re out aboard a ship sailing in the oceans, which cover 71% of the surface of our planet. Which geometry would be useful to you for the purposes of navigation? Spherical geometry, of course. True, spherical geometry would stop being useful as soon as you disembarked and tried to climb a mountain or descend a crater (although it would still apply on the surface of a large desert); but to navigate over that 71% of our planet you wouldn’t need any corrections to your basic spherical geometry, at least not over the oceans; which is analogous to plain hyperbolic geometry on a large scale in our universe (the observable part of it). And hyperbolic geometry is the mathematical background of special relativity. However, some objections might still remain in the mind of the reader. In special relativity there is no absolute frame of reference! An observer A moving with respect to observer B is free to declare A’s frame of reference “stationary” and B’s frame moving; but B, too, has “equal rights” with A to declare B’s frame of reference “stationary” and A’s the one that is moving. After all, that’s what relativity is all about. But if the previous diagrams depict the geometry of the universe, then there seems to exist an absolute frame of reference: it is the one defined by the axes s, t and the asymptotes! How is special relativity compatible with an absolute (or “preferred”) frame of reference? Relativity or absolutism? The conceptual problem arises from the fact that special relativity has always been seen only as a useful stepping stone in approaching “the real thing”, i.e., general relativity. The latter requires knowledge of calculus, and of topics that even professors of mathematics do not usually study deeply, unless they have an interest in physics; whereas the former requires only knowledge of high-school math — square roots and imaginary-complex numbers is as far as one needs to go to master special relativity. You would lack the need for an absolute frame of reference if you were out on the surface of the ocean sailing aboard a ship (see my previous analogy) and were only concerned with describing events that happened on either your or other, passerby ships. That’s fine, your relative spherical-geometry calculations would be correct and you wouldn’t need an absolute frame of reference. But that doesn’t preclude, in and of itself, the existence of an absolute frame of reference, which is the ocean seabed and the faraway lands that you cannot see even with your binoculars; objects that, all together, form the nearly spherical surface of our planet. That’s what most books on special relativity do: they tell us how to navigate in the ocean based solely on the formulas of special relativity, ignoring the absolute frame of reference (the faraway lands and the ocean seabed), which however exists, but at great distances. Ditto for the universe on a grand scale. Using special relativity locally, we lack the need for an absolute frame of reference and we know that the theory will break down in the vicinity of a massive object. But this is not evidence for the nonexistence of an absolute frame of reference, which is defined by the faraway stars, galaxies, and galactic clusters and super-clusters in the universe. They are there. When you move from our galaxy to the Andromeda galaxy, you move with respect to them. And it is you who ages slower than your twin who remains on Earth, not your twin who takes the entire Milky-Way-plus-Andromeda system (plus all the other galaxies of our local group, plus our local super-group, etc.) and moves away from you — something outrageous and in total conflict with reality. At this point I’d like to make some observations related to the well-known “Twins Paradox”, which, I believe, is presented from a wrong perspective in the chapters of some books on special relativity. The Paradox of the Paradoxical Authors Writing on “Twins Paradox”For decades, books upon books have been written telling us that one of the twins who travels at speed close to the speed of light will age less than her brother who stays on Earth; and that this will be observed when the travelling twin returns to Earth, at which time the difference in age between the two will be verified by mere observation: the traveler will be young, the “lazy” one old. Up to this point all is fine. From this point on, however, some books (and in particular some popularizing ones on relativity), unreasonably enough, tie themselves into the following conceptual knot: Since, according to relativity, there is not an absolute (but only a relative) frame of reference, the traveling twin, call her A, may consider her frame of reference stationary, and that the one who travels is her twin brother, call him B. The formulas of relativity are symmetric with respect to A and B, therefore they can apply equally, either from A’s perspective or from B’s perspective. That’s why we talk about “relativity” after all, correct? From there on, the books that I refer to start the project of proving to us that, in spite of the symmetry in special relativity, only the traveler twin (i.e., A) will remain younger in reality, whereas B will age more than A. But how can that be? If there is full equivalence among the two frames of reference (A’s and B’s) in special relativity (which, there definitely is), then whatever logic we follow considering B as stationary and concluding that A ages less than B, can be followed in an exactly symmetric way considering A as stationary and concluding that B ages less than A. How can it be that only A ages less, given the equivalence? We can’t have the pie whole and eat it. The authors of such books do the calculations from B’s point of view and prove (as the numbers show) that A will appear younger than B upon return. But when they do their calculations from A’s point of view (which assume that A stays put) they indirectly insert in them the assumption that A is moving. A typical such example is Paul Davies’s “About Time: Einstein’s Unfinished Revolution” (1995, Simon & Schuster). Other authors, without doing the correct calculations (which would prove them wrong) claim that the asymmetry between the twins is due to “the accelerations”. That is, since A is traveling, in order to return to Earth she must decelerate, change course, and subsequently accelerate, reaching the same constant speed as before, but this time heading toward her twin brother B who’s waiting on Earth. There, they say, i.e. in the deceleration/acceleration, is where symmetry is broken, because only A will feel the deceleration and acceleration (and conclude, objectively, that she has been traveling), whereas B will feel nothing at all. But it is not necessary that there be decelerations/accelerations! Simply, A may have communicated in advance with a third person A΄ (the third member of a triplet, say), so that the moment A reaches her destination of the outgoing leg of the trip she meets A΄, who has already started the trip a short while ago, and has already acquired A’s speed in absolute value but moves in the opposite direction. A and A΄ synchronize their clocks the moment they meet each other (since, momentarily, they share the same system of reference), and A΄ continues moving toward the earthbound B (whereas A retires and goes for a coffee — whatever). Thus, A΄ simulates fully an A that arrived and turned back, and did that instantaneously, without any deceleration or acceleration. I omitted mentioning that the real age of A΄ is of no concern to us (let A΄ not be a member of a triplet). What matters is what interval of time will have been recorded on the clock of A΄ upon reaching Earth. Needless to say, the clock of A΄ will be showing a shorter time than the one that B will observe on his own clock. How is the asymmetry now explained without decelerations/accelerations? Simple: twin A is the one who really travels! She travels with respect to the absolute frame of reference defined by the spacetime of stars, galaxies, and other heavenly bodies. That’s why B ages more than A. The trip is not relative but absolute. The same thing is shown to us by the muons that reach the sea level — an observation that has been used time and again in support of the correctness of special relativity. The observation is correct: the average muon “ages” much less than what we’d expect had it not moved at a speed approaching the speed of light; that’s why it manages to reach the sea level before decaying (on average, of course). But given the equivalence of the special-relativistic formulas, the muon might consider itself stationary and the Earth coming to meet it at a speed near the speed of light. However, in that case Earth should remain younger than the Earth that the muon actually meets. To see that this cannot be the case, and because the muon’s journey from the upper layers of the atmosphere to the surface of our planet is too short to figure out how much the Earth “aged”, we should see the same example but by means of a trip that takes place over much longer distances and durations. So let’s imagine the following: Suppose that, 3 million years ago, A started a journey from a planet at the remote-from-us outskirts of the Andromeda galaxy, exactly three million light years away from Earth, heading toward Earth at speed v = 0.9999999998c. (We assume zero acceleration for A, as this journey had started a little earlier, so that A reached the desired speed v exactly three million light years away from us and three million years ago.) At that time, there were no human beings on Earth yet, but only humanoids of various species of australopithecines. Let’s calculate the time that A will experience during this epic journey. According to the special-relativistic formula, the time t´ of the trip as experienced by A is: t´ = (1 – α) t / √(1 – α2), where α = v/c = 0.9999999998. Plugging in the value of t = 3,000,000 years, we get t´ = 30 years. That’s how long the trip will last from A’s perspective, although 3,000,000 years will have passed on Earth, assuming it “stationary” (or “barely moving”, because our Milky Way doesn’t make any discernible motion with respect to the Andromeda galaxy within only 3,000,000 years). But, within those 3,000,000 years, the australopithecines will have evolved to our own species. They will have become we. Therefore A, upon arriving at Earth, will not greet australopithecines but us. Obviously we’ll welcome A with a very different ceremony (bands playing, speeches, exchanges of gifts, etc.) than that of the australopithecines (who, most probably, couldn’t care less). But now if, according to the full equivalence of the frames of reference in special relativity, we consider A stationary and the australopithecines “coming” toward A at a speed of v = 0.9999999998c (together with the Earth, the entire Milky Way, and with A leaving the Andromeda galaxy behind — that outrageous situation implied by the symmetry of special relativity) then we’ll conclude from the same formula as above that the journey of the australopithecines will last for t´´ = 30 years from their perspective. Therefore, that australopithecus who was a baby when the journey of the Earth-etc. started, is now simply a grandpa and ready to kick the bucket after 30 years, but is still alive when A meets him! So what will A meet? Human beings of our kind, or australopithecines? One needs not be a rocket scientist to conclude the obvious: A will meet us, Homo sapiens. And what will A be seeing observing Earth with a super-telescope for the duration of the journey? A will witness, within 30 years and in super-fast-forward motion, the entire evolution of humanoids, from the Australopithecus era to our own one, the era of Homo sapiens. (Let’s ignore the fact that, due to the tremendous speed of A’s spaceship, all photons that reach A’s telescope from the “front”, i.e., from Earth, will have been converted to gamma rays, due to the Doppler effect. We said we’ll make a thought experiment, okay? Not a realistic simulation!) Without the rest of the universe to provide a fixed frame of reference, both the traveling (with respect to it) observer and the stationary one will do exactly the same calculations because their situations are symmetric. Ergo — always assuming the rest of the universe absent — we don’t have any meaningful way to distinguish between the two and predict which of the two mutually conflicting events will take place. It is only when we add the rest of the universe in the picture (and do not use just bare-bones relativistic formulas) that we may observe (not just compute, but observe, as we do with the muons) that A will age by only 30 years during such an epic 3,000,000-year-long journey. But by adding the rest of the universe into the picture we do nothing short of adding a fixed frame of reference, which is the real thing, the absolute frame of reference that special relativity took pains to eliminate. There is nothing wrong with the spacetime of our universe serving as an absolute frame of reference. The confusion has historical roots and causes. Newton’s framework assumed an absolute space and an absolute time. Then came Einstein, who was influenced by Mach’s earlier philosophical (and relativistic) view. In 1905–1906 Einstein said that there is no absolute time: time is relative to the observer’s motion. Also, there is no absolute space; lengths of space are also relative to the observer’s motion. Correct. But there is an absolute spacetime, which is defined by the distribution of matter in the universe. This was not said explicitly by Einstein in 1905–1906, because it wasn’t necessary. And since relativity became a buzzword in 20th century physics, nobody wanted to spoil the revolution that relativistic thought brought to the world of physics by referring to an absolute frame of spacetime. Overall conclusion: special relativity describes the hyperbolic absolute spacetime of our universe to a great approximation, corrected here and there by the more accurate general relativity, which is necessary only at the really-really rare (at a universal scale) places where truly massive objects exist.
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If you’re a mathematician, you
have absolutely no trouble understanding the notion: “whenever the
dimension of time is considered in conjunction with any (or all) of
the other three [dimensions] the geometry of the
resulting space-time is hyperbolic”.
Every straight line with slope
smaller (less steep) than that of the asymptote corresponds to an
entity
moving at a speed less than the CSL (slower than the speed
of light).
