The Book Of THoTH

Here's a brief (stretching the definition of "brief"), simple introduction to special relativity I typed up for something else a few months back. Might help some browsers of this forum.

Before 1905

The story began four centuries ago with Galileo Galilei. Galilean relativity said that the basic laws of physics are the same in all inertial reference frames (an inertial reference frame is one in which there’s no acceleration—there’s no change in speed or direction); this is called the relativity principle. Galileo’s thought experiment involved a sailor dropping a knife from the top of the mast of a moving ship. Someone in the water watching as the ship goes by would see the falling knife follow a curved path because, while gravity is pulling it downwards, it also has a horizontal (forward) velocity—that of the ship. The sailor (and anyone on the ship), however, sees the knife fall from the top of the mast to the bottom; it’s as if the ship were sitting still (they all, like the knife, share the ship’s forward velocity so they don’t notice it).

So the laws of physics appeared to work exactly the same in every inertial reference frame (whether "moving" or "stationary"); playing ping pong in your basement is no different than playing on an airplane that isn’t accelerating. These early forms of relativity assumed that the lengths of objects and the rate at which time passes were the same in all inertial reference frames as well.

Then in the late nineteenth century electricity and magnetism were unified and described by Maxwell’s famous equations which revealed that light itself is nothing but an electromagnetic wave. The equations predicted that light would move at a certain speed, c, or about 3.00 x 10^8 m/s. But in which reference frame did light move at precisely this speed? *Surely if one moved toward a light source at half the speed of light he would measure the speed of light to be 3/2 c (the regular speed of light plus his own speed) and if he moved away from a light source at ½ c he would measure the speed of light to be only ½ c (the regular speed of light minus his own speed). So it was thought that there must be some special reference frame in which one measures the speed of light to be exactly the one predicted by theory. It was also believed that since light is a wave it must propagate through some medium (like water waves through water or a wave on a string)—they called this the ether. The speed of light predicted by Maxwell was thought to be with respect to this ether.*

Sadly this didn’t satisfy the relativity principle. Maxwell’s equations were different in different inertial frames. The laws of electricity and magnetism seemed to change with the reference frame. There was a state of absolute rest, a special inertial frame: that of the ether. An experiment to measure the earth’s speed relative to this ether was cooked up by Albert Michelson (the first American to win the Nobel Prize in Physics and inventor of the Michelson interferometer, still widely used today—in gravity wave detectors, for example) of Cleveland’s Case School of Applied Science and Edwin Morley of Cleveland’s Western Reserve University (the two merged in 1967 to form Case Western Reserve University). They essentially measured the speed of light in two perpendicular directions; it makes sense that if such an ether exists the measured speed of light would depend on the direction it traveled. The result was puzzling—the light seemed to travel at the same speed in both directions.

Enter Einstein

In 1905 a lowly patent clerk (but one well-trained in physics—crackpots not trained in physics often gloss over that point in their bid for martyrdom) suggested that the principle of relativity does hold for electromagnetism. He advanced the theory today known as the special theory of relativity. And it was a doozy. Let’s take a look at it.

The theory is based on two postulates. The first is the familiar principle of relativity: the laws of physics are the same in all inertial reference frames. The second concerns the constancy of the speed of light: it says that light moves through empty space at a speed, c, that’s independent of the speed of the source or the observer.

That second postulate is often difficult for people to swallow. It says that no matter how fast you’re going, no matter if you’re heading toward or away from a light source, you’ll always measure the same value for the speed of light. Drive at 99.999999% of the speed of light and turn on the headlights--you still measure the light to race away from you at the standard speed of light. So the ideas above between the red asterisks are incorrect. Those were commonsense, intuitive notions; but the universe doesn’t work that way.

It’s important to realize that motion and speed are relative—those words only mean something when speaking of some reference point. It turns out there is no state of absolute rest, no ether. When you’re going "35 miles per hour" in your car that’s relative to the ground. Relative to the guy behind you who’s doing 30mph you’re only going 5 mph. So out in deep space what does it mean when Captain Picard gives the order for all stop? I imagine they have some reference point when they say that (or bad writers). But it’s important to understand that below when I say something like "he’s moving at 99% of the speed of light" that only means relative to some other observer (this (and any) mover can consider himself to be at rest--this helps in understanding why everyone measures the speed of light to have the same value).

The consequences

Now I’m going to tell you some things without really showing how or why they come about (though I’ll try when it's possible). That will keep this simple without messy derivations and the harder parts of the theory getting in the way.

Three of the most basic mechanical quantities are affected by this: time, length, and mass. However, the effect only becomes noticeable at relativistic (meaning close to the speed of light) speeds, which is why nobody hit on this theory until the beginning of the 20th century.

Time dilation

It turns out that at speeds close to the speed of light (once again this is relative to someone else) time itself slows down. This means that if you stayed here on the earth and I sped off at 75% of the speed of light you would (assuming you had an amazingly powerful telescope to peer into my spacecraft and see my clocks) observe my clocks as taking longer to tick off one second than your own. This means that I would age more slowly than you and all that good stuff. But remember, this is relativity. Since I’m moving in an inertial reference frame I could easily say that I’m at rest and that you're the one moving away from me at 75% of the speed of light. Each observer sees the other’s clock as running slowly. Very symmetric.

A more mathematical look (and the simple explanation as to how it follows from special relativity) isn't too tough to produce or follow but it's a little peripheral to this first post.

Length contraction

It turns out that length contracts when you’re moving very fast (along the direction of motion). So when I buzz you (who is "stationary") going 75% of c you’ll measure my spacecraft to be shorter than I would measure it to be. But it isn’t just the length of the craft that shrinks but distance itself. An example using what we learned above about time dilation will help (after these visuals stolen from here).

Spaceship Moving at the 10 % the Speed of Light


Spaceship Moving at the 86.5 % the Speed of Light


Spaceship Moving at the 99 % the Speed of Light


Spaceship Moving at the 99.99 % the Speed of Light


The third closest star to the sun is called Wolf 359 (Trek fans will recognize the name as the site of the first battle between Starfleet and the Borg in which Starfleet had its balls handed to it) and it’s 7.8 light years away. That means that at the speed of light it would take 7.8 years to get there. Moving at 90% of the speed of light it should take about 8.7 years to get there. Makes sense, doesn’t it? But wait: 8.7 years as measured by observers back on earth is much longer than what our relativistic astronaut will measure (time slows down for him). After 2 years and 4 months pass for the people back on earth, the astronaut will just be reaching the one year mark on his calendar. So when 8.7 years have passed for the people of earth and they know their astronaut must surely have reached Wolf 359 he’ll have experienced not quite 3 years and 10 months. So what can explain the fact that he completed what should have been an 8.7 year trip in less than half the time? The distance to Wolf 359 must have contracted from the astronaut’s point of view. That explains how he could travel so far in such a short time—the distance itself shrank.

Again, the situation is symmetric. The people "at rest" on earth see the spacecraft shrink with respect to the distance (which they see as unchanged), whereas the astronaut sees his spacecraft’s length as unchanged but the distance to Wolf 359 is contracted (by the same factor as the people on earth see the length of his spacecraft contracted). Neither side is "wrong."

Mass

This is a tricky one. You could say that when moving at speeds close to c (again, as viewed by someone "at rest") an object’s mass increases. Mass is essentially a measure of how difficult it is to accelerate an object; so if objects become more massive the closer they get to the speed of light then it becomes more and more difficult to accelerate them further. Which is why c is the ultimate speed limit and nothing with mass could ever reach it.

Now if you were to go deeper into special relativity you would discover that this concept is not often used anymore (though it’s a good way for those new to the theory to visualize the "c as a cosmic speed limit" concept). Energy and momentum are the quantities that are really considered here. Einstein himself said:



So that’s that.

You may be wondering where these come from or, alternatively, you may have heard of something called the "Lorentz transformation equations." We’ve been talking about how different observers see different things and yet neither is incorrect; suppose we have two observers, each defining his own coordinate system. The Lorentz transformations are how one would convert from one coordinate system to the other. It is in this conversion that we discover the above consequences of the theory (for example, that time is measured differently by the two observers in the different coordinate systems); they are the connection between those consequences and the original two postulates of the theory we mentioned above. But we’re avoiding the math here which means we must avoid the transformations themselves.

Two more important consequences

There are two more extremely important consequences of the theory that came later (i.e. were not a part of the original paper). One is directly attributable to Einstein, the other is not.

E=mc^2

Perhaps the most famous equation in physics. It gives the speed of light squared as the conversion factor between mass and energy. It shows that mass can be converted into energy (as in when matter and antimatter annihilate leaving behind only photons) and vice versa (as in pair production—high energy photons "split" into a matter-antimatter pair). Very important to nuclear physics as well.

The actual equation for relativistic energy is: E^2 = p^2 c^2 + m^2 c^4. When an object is considered to be at rest (i.e. when you’re in the same inertial reference frame as it is) then the velocity is zero and the momentum, p (m*v) is 0. It then reduces to the familiar equation.

Spacetime

This one isn’t Einstein’s. A former professor of his named Minkowski is the one who unified space and time. This can help you visualize why time slows down (as seen by others) for you as you approach the speed of light—the more you move through space, the less you move through time. A bit crude but it's helpful. This also ensures that units of distance can function as units of time and vice versa. For example, a second can be used as a unit of distance—that which a beam of light traverses in one second (about 3.00 x 10^8 meters as we said above). This idea is behind the notion of time as the fourth dimension: it’s simply a component of 4-dimensional spacetime.

The light cone

The light cone represents the points in spacetime which information about an event can reach, taking into account the c speed limit (it's impossible graph all the spatial axes along with time so they always drop a few). Anything outside the cone is inaccessible. The cones are formed by beams of light; they’re at a 45° angle to show that light travels through just as much space as it does time (it’s said to be lightlike). Events that are separated such that some material object could travel between them are said to be timelike. Spacelike events are those that even light can’t connect.



Concluding thoughts

That's the bare bones of special relativity. I've left out some notable concepts like interval but I think this is enough to give anyone an idea of what's being talked about when special relativity is discussed. The lack of images couldn't really be helped, though I think the few I put in worked.

All of the predictions of special relativity have been thoroughly tested experimentally and the theory had to be taken into account for quantum mechanics to work correctly. Recently there have been suggestions that revisions (more like additions) should be made to the theory to take into account certain things.

Follow that?

Thanks Penthar, it's easy to get lost with that stuff, its as well to have something like this as reference.

Well done

That was really helpful and did explain a lot of things, but my line of reasoning doesn’t always work the same as other peoples so I have lot of questions. It’s probably best if I post them one area one at a time to avoid confusion.

One of the things that made sense but kinda created more questions was the fact that objects gain more mass the nearer they get to light speed. I understand that. Then it occurred to me that light must have mass because it moves and momentum can’t exist without mass. If light can reach lightspeed why not other things too? I’m assuming that light must have less mass than other things, does that mean nothing has less mass than light? If something has less mass than light then it would or could also exceed light speed couldn’t it?
Another question, is the mass of light constant, or put it another way…..before something reaches light speed, it has to be slower than light speed or even completely at rest, so light is an end product…of what? What was it before it was light? Hmmmm The sun is light…or emits light…why does it do that?
Is light speed still considered to be constant, (I suppose that relates to the mass question, doesn’t it?) I’m sure I’ve read that that’s not the case, but I don’t know if I’ve picked that up from one of those pseudo physics sites/books that ummmm…. might be not quite correct.

So I get confused. Does the mass of light never change? Or does it change and that’s why light speed can’t be exceeded (ie after that the mass is too great to move faster).

Apologies if the questions are a bit muddled, I know what I don’t understand but it’s a little difficult explaining my queries.
Wait till I start on the spaceship stuff you think that lot was confusing.

Isis

Ok, first question of the day.



That's not exactly true. Back in the day (Newton's day, that is) when momentum was thought to just be mass x velocity that would've seemed to be true. But remember that mass and energy are equivalent (E=mc^2). So what happens if we rearrange that equation to solve for mass and plug that into the tried-and-true Newtonian equation for momentum?

Well, we see that m=E/c^2. Mass equals energy divided by the speed of light squared. So instead of p=mv we have p=(E/c^2)v. What's the velocity v when we're dealing with light? It's c--they don't call it "the speed of light" for nothing. So in the end you see that momentum p is equal to energy divided by the speed of light: p=E/c

The moral: light may not have mass but it's got energy and that's all it needs to have momentum.



Well, there are some theoretical ideas for cheating to move faster than the speed of light (I say "cheating" because even in those scenarios no one is technically moving faster than the speed of light). But nothing with mass can accelerate up to and pass the speed of light.

Light has no mass. Put slightly different, it has no rest mass--that's ok because it's never at rest! To have less mass than that means you've got negative mass. And even that wouldn't mean something's moving faster than the speed of light. It would take some kind of "imaginary mass" to do that.

The reason lies in the fact that there's a factor that pops up all over special relativity that has the term sqrt(1-v^2/c^2) in it (I know that looks ugly--it's the format, I swear). As long as the velocity v is less than the speed of light c you have 1 minus some fraction smaller than 1 in those parentheses. But as soon as the velocity is bigger than the speed of light you have 1 minus some number bigger than 1--that is, a negative number. What's the square root of a negative number? Those are called imaginary (or "complex") numbers. But how does one physically interpret an "imaginary" mass?

Interesting side note: there was a paper in the American Journal of Physics 8 years ago called "Complex speeds and relativity" in which it was shown (for fun) how using imaginary speeds one can get around the light barrier. But the author admitted that she had no idea what that would mean physically or how you'd go about doing that. She later used the idea to write sci-fi. Here's the abstract:

"The quest to find faster-than-light particles has intrigued physicists for decades, though it has yet to turn up any real candidates. Even if a superluminal universe does exist, we have no way to reach it given that we must go through the speed of light, which to the best of our knowledge is impossible. In this paper, I show that by making speed complex, we can go around the speed of light in a manner analogous to the way a car faced with an infinitely tall road block might leave the road to go around that barrier. The treatment is a mathematical device; no known physical interpretation exists for the imaginary part of a complex speed. However, it can provide an entertaining problem in special relativity, one that may encourage students to think about the connections between equations and the physical universe."



Well, the answer to the first part is yes and no. The mass of light is constant--it's zero! However, the energy of light can vary considerably so if you consider (as chemists do) light to have some sort of effective mass just because it has momentum (which depends on its energy) then, sure, that "mass" is different from photon to photon. But that can get confusing.

Now you said "reaches light speed." Nothing ever reaches light speed. It's either there or it isn't. Things with mass travel slower than the speed of light. And things without mass travel at exactly that speed--no faster, no slower. Light always travels at exactly the speed of light. It can't accelerate at all. We've all heard that light travels slower in water or glass than it does in a vacuum but that's not precisely true. What really happens is that light is being continually absorbed and reemitted by the atoms making up such media--the net effect is that there's a slight time lag in going from point A to point B that you wouldn't get in a vacuum. But at no point does a photon (light particle) travel slower than the speed of light (299,792,458 m/s).

I'm not sure what you mean when you ask what light was before it was light. Photons are created (among other ways) when an electron in an atom falls down an energy level--that is, it loses energy. The exact amount of energy lost by the electron is the energy carried away by the photon. Hot things (the sun, the stove, etc) emit visible light because the electrons in them are excited and jiggling like crazy. The photons coming from them are simply them releasing some of this energy.



Yes, it's still widely thought to be a constant. However, there are two major twists to the story.

The first is that certain observations made over the past few years seemed to suggest the possibility that the speed of light may have changed in the past (there was a big hubbub at the online sources a year or two ago following a communication Paul Davies wrote to Nature on the subject). Anyway, it appears that these observations might've been incorrect: see the "A Real Constant After All?" thread in this forum. It's only a few threads below this one.

The second wrinkle is the recent wave of "VSL cosmologies"--these are cosmologies based on a varying speed of light. They're championed as alternatives to inflationary cosmologies (which we've discussed) and they attempt to solve all the problems with the original big bang theory that inflation resolves. VSL cosmologies are based on the notion that the speed of light was much faster in the opening moments of the universe's existence than it is now. But, as far as I know, even in those theories the speed of light is regarded as being constant now.

P.S. Some of these questions have a lot of background behind them and sometimes I'll go on, forgetting that you may not have a crucial piece of the background puzzle. So stop me if that happens and we'll make sure to cover it.

If light doesn't have mass then that makes things a whole lot clearer. Rearrranging the equation made sense. Am I understanding this correct, energy and mass are two different forms of the same thing so everything consists of either mass or energy.... and light is made up of energy.



The idea of imaginary mass really appealled to me (yeah, I'm sad!) So I looked it up and found this...



Does that sound right? It sounds interesting. If that's correct we could never exceed lightspeed without moving from the 'real' mass universe to the 'imaginary' mass universe.


That I don't understand. I understand the statement obviously but there must be an acceleration process of some kind to get to light speed even if it's a fraction of a nanosecond or something it must have to accelerate to light speed. Maybe it's not measurable, but it must do.

Another thought that I had when reading your reply was what about coloured light, all coloured light will move at exactly the speed of light but why, when different light has different wavelenths. The different wavelengths make the colors disperse differently so why doesn't that affect the speed at which it travels?


Yep, I did read that at the time, but I'll reread it.


How do they know that? That's kinda confusing if the theory about there being two different universes, one with real mass and the other with imaginary mass is correct. Or maybe it fits and that's how the two universes were created.



I'm following okay so far but if I do get lost I'll let you know, thanks.

Isis

Right. But be careful with the "made up of" bit. Things have mass and they have energy. You wouldn't say that you're made of mass.



Well, it's science fiction. You're still left with the real-world problem that imaginary masses and energy don't seem to have any physical interpretation.



Light doesn't have to get up to the speed of light--it's created traveling at that speed and it's destroyed traveling at that speed. It never accelerates.



This one's tricky. First, in a medium like glass the different wavelengths of light are absorbed and reemitted at slightly different rates. The end result is that some wavelengths seemed to be slowed down more than others. That's how a prism breaks white light up into the whole spectrum of colors. In a vacuum, though, the different wavelengths should all travel at the same speed. That's just the way light is.

I said "should" because certain forms of a new version of special relativity being developed right now ("doubly special relativity") predict that in some circumstances the speed of a photon does change depending on its wavelength. But that story's still unfolding.



At the moment they don't.

Science fiction can be based on science fact though, can’t it – Kinda like the person you mentioned earlier who theorised about imaginary speeds to get around the light barrier and then wrote a science fiction book. A lot of science seems to start with a theory that then gets proven (or not) in due course. That’s why I wondered about there being two universes one where the speed of light cannot be exceeded and one where is constantly exceeded.

Time dilation I wondered about for years, but I still don’t quite ‘get it’



Symmetry always interests me; I need to think about symmetry some more it kinda strikes me as being important in the big scheme of things, but I haven’t figured out how yet. (I understand it in relation to this thread though.)

Getting back to the time dilation issue, does this mean that time dilation is just an illusion if neither party is aware of it themselves but perceives the other as being subject to it? I ask because later when you later talk about it taking 8.7 light years to get to Wolf 359 I kinda got the impression that if a person spent all their life flying around the universe at light speed physically experiences time slowing down. For example they might live for the Earth equivalent of say 150 years instead of the 75 years they would have lived had they been resident on Earth. I’ve misunderstood somewhere but I’m not sure where.

The other thing that I didn’t grasp is the length contraction thing. I understand that the contraction is relative, so that the person on the spaceship wouldn’t experience any signs of the contraction themselves. Length contraction is an illusion isn’t it – or maybe more of a distortion of reality, it doesn’t physically happen, that’s what I mean.

Anyhow, you can see why I’m getting confused, one minute I’m thinking it’s an illusion but then the light years to Wolf 359 makes me think its not an illusion.

Once I understand whether I’m looking at a physical reality (things really DO slow down / contract at light speed) or an illusion (things APPEAR to slow down / contract at light speed) then I can maybe understand WHY it happens. I’m halfway there with the symmetry and it being a relative concept but I’m missing something vital.

Isis

Sure, but sci-fi tends to go far beyond what's known and possible, which is why it's important to figure out where that line is whenever you're reading/watching it.



Not at all. Both side's perspectives are equally valid--there is no "correct" time, etc. That's the relativity part of it all. Your time need not be the same as mine.



Near light speed, remember. Never at. And light years are units of distance. Nitpicks done.

Anyway, time only slows down for the astronaut from the point of view of someone "at rest" (like someone back on earth). The astronaut doesn't notice anything different at all--time seems to tick by normally for him, as always. Mechanical clocks, biological clocks, whatever all slow down but only from the point of view of the observers at rest. So he'll still die in 75 of his years. But that might be a lot more years for the person back home on earth.




Yes, it physically happens. The distance to Wolf 359 really shrinks for the astronaut on the way and the length of the ship really shrinks for someone back at home on earth watching. Again, both perspectives are equally valid. Length simply isn't absolute.*




This is why they say you have to abandon common sense and accept some fairly strange things in modern physics. This isn't like a straw in a glass of water appearing to be bent, this is all very real. But it all depends on your point of view. It's relative.



Well, there is a pretty painless derivation that can be done involving a simple thought experiment. The problem is that it's (obviously) mathematical (though it really only uses algebra) and I don't know how well it would look in this format (no superscripts, etc). But it does help you to see how and why things can physically be that way.

Thanks Penthar, I understand that now.

The last part that got me thinking was this...



Can we look at light cones in a little more depth, if that’s okay? I seem to have come across them a lot in recent years so maybe it’s time I understood them a little better. A light cone in formed by two intersecting 45 degree angles,yes? The exact point of intersection is ‘now’ and the two sides (or cones) represent past and future, yes? I can see the symmetry there similar to the earlier example. Light cones are theoretical devices not actual physical phenomena aren’t they? But what if they could be artificially created I thought.

If you took a light cone and filled it with exotic matter, in theory would that create a wormhole?

Going off on a tangent (no pun intended) the 45 degree angles make me think of UFOs. Don’t ufo’s often make either two 45 degree turns or one 90 degree turn before disappearing? That 45 degree angle and the way UFO’s just disappear is interesting to consider in relation to light cones, worm holes and the like.

Another thing that occurred to me was why do we assume there is just one light cone? This kinda fits in with my choice of avatar, which is a series of light cones and represents the paths out of the universe. Supposing there were a series of light cones not just one. What would that make the pivot point? (present) Surely it could be a big galactic intersection. I know that's all conjecture but I get the impression that conjecture is used in physics, I'm just seeing how things fit together.

I wondered about this… you said that light travels through just as much space as it does time (in the cone), is this a completely separate rule to earlier when you said that time dilates and space contracts? (on the way to Wolf359). Is that because at the axis ('present') the rules are different than either side of the axis? Does ‘present’ have a different rule to ‘future’ and ‘past’?

Isis

Light cones show you graphically which things in the past/future of an event can influence/be influenced by it. If something is inside the light cone it means that a signal could pass between it and the event in question without anything having to go faster than the speed of light. If the sun explodes right now it can't influence the writing of this post--it'd take 8 minutes for the influence to reach me and I'll be done by that time.

Light cones aren't physical objects so we couldn't put anything in them or create new ones. They're just a way of representing graphically the causality of spacetime--which causes can have which effects when you take into account the light-as-an-absolute-speed-limit rule.



First of all, this is why light is generally drawn at a 45 degree angle so this should serve to answer what you said before about that.

Light travels one light-second in one second. That's just a definition. All the things you and I know travel significantly less than a light-second in one second--maybe you'll go a few tens of meters in one second in a fast car (a light-second is 299,792,458 meters, by the way). We travel through more time than we do space. The faster we go, obviously the more space we'll be covering (if you're sitting still, you're not traveling through any space at all, just time). The key is that there's a quantity called the interval which involves the three spatial dimensions and the one of time. And it's seen as being the same for everybody.

I'm sure you know the 2-D Pythagorean theorem from Euclidean geometry: a^2 + b^2 = c^2 or, alternatively, c = sqrt(a^2 + b^2). The interval is kind of like that, except (as I said) it involves a few more dimensions.



Events connected by a beam of light (like the sun emitting photons 8 minutes ago and your eye absorbing them right now) have an interval of zero because light covers just as much space as it does time.

The dilation of time and contraction of space (along the direction of motion) for some object moving slower than the speed of light is related to this in that, since the interval is seen as having the same value by all observers, a longer time must be counteracted by a smaller length to keep the final value the same. Just another indication that time dilation and length contraction accompany one another.