if the faster you go, the slower time becomes, doesn’t that just mean that you are going at normal speed?

No, because there is no such thing as a “normal speed.”

“Normal” implies a priviledged frame of reference.

Yes, that is the whole point of (general) relativity. As far as you are concerned, you are standing still,everything else is moving, the other places see time differently from you, and if you measure the speed of light from your point of view, it still travels at the same speed as anyone else who measures it.

Where you notice issues is “special relativity” which applies when you accelerate or decelerate. If you are just coasting, then you measure things, the guy whizzing by sees time slower. From his perspective, he sees your time going slower. Whoever changes speed so that you can meet up and compare notes while see odd things happening so that the percieved times can sync up.

If have the opposite question: Suppose I leave earth towards a star that is ten light years away. Also suppose that I am going about 90% of the speed of light, so it takes me about 11 years to get there, as measured by someone at relative rest. But because of my velocity, I experience time more slowly – by a factor of about 2.3 – and so it *feels to me* like I arrived that that star in less than 5 years. Am I right?

Yes. Special relativity can be interpreted as saying we’re all moving at the speed of light through the time dimension. As we move relative to other objects in space, we are diverting some of that velocity from the time dimension to the three spatial dimensions, thereby speeding us up in space and slowing us down in time.

That’s my pop sci understanding, at least. A real physicist will be along shortly to clarify and/or correct me.

Yes, but from your POV you are experiencing time as usual, and the reason you age less than 5 years is that the span of distance has been foreshortened by a factor of about 2.3.

What bothers me about my scenario is the fact that I appeared, by my own measurements, to have gone faster than light.

Is there a way, within relativity, to say that the initial measurement of ten light-years was faulty, or dependent on a particular frame of reference? In other words, Should I consider that when I went from earth to that star, I was actually at rest while the star approached me at 0.9C, and the earth receded from me at 0.9C, and the true distance (in this frame of reference) was only 5 light years?

If a person who’s sitting on Earth measured the distance to be 5 light years, then the person moving toward the star at 0.9 c will measure the distance to be less than 5 light years. (I should say, who’s moving towards it at 0.9 c *in the Earth frame*. You’re *always* at rest in your own reference frame.)

This is how special relativity works: everyone (regardless of reference frame) finds the speed of light to be 300,000 km/s relative to them, but people in different reference frames find distances and time intervals to be different, and different in just the right way that it accomodates the fact that all of them are measuring the same speed of light (contrary to our Galilean intuition that you should measure a different speed of light depending on how fast you’re going).

Yes, this is exactly it. When you say the star is 10ly away, you mean “the star is 10ly away in a frame of reference that is at rest with respect to the star (and also presumably the observer watching you take off on earth)”.

In a frame of reference that is moving at 0.9c between earth and the star, the star is not 10ly away ever (see Lorentz–FitzGerald contraction).

Exactly. Distance depends on your frame of reference. From your perspective Earth and the star are 5 light years apart. From the perspective of someone on Earth, the star is 11 light years away. Neither measurement is more “correct” than the other.

From your perspective the universe is moving at 0.9C relative to you. So the entire universe is Lorentz-contracted in the direction of motion, and your clock is moving faster than everyone else’s clocks.

But from the perspective of the universe, your spaceship is moving 0.9C. Your spaceship is Lorentz-contracted, and your clock is moving slower than everyone else’s clocks.

People above are giving different answers because they seem to be addressing different aspects of the problem. Let me make a stab at putting them all together.

In all reference frames you perceive time passing at the rate of 1 second per second. It does not matter at all what speed you are traveling or whether you are undergoing acceleration.

If you travel to the nearest star under an acceleration of one gravity you will feel you have aged normally every second of the time. All your instruments will show this. Everything that would normally rot or decay in x amount of time will rot or decay in x amount of ship’s time. Nothing ever stands still; time always passes.

But, and here’s the big but… The measurement of the amount of time will not be agreed upon by people in other reference frames. I don’t remember the exact numbers off the top of my head, so I’ll make some up. If you travel to Alpha Centauri and back, it will feel to you and everything in your ship and reference frame that two years has passed. Clocks on Earth will say that 11 years have passed. Both are completely right. You can never say that you have gone faster than the speed of light, because when you measure it you use relativistic equations that include your speed. You are measuring spacetime as one entity, not space and time as separate entities. The effect is that the distance has shortened sufficiently so that that rate is always less than the speed of light.

This is what drives people crazy about relativity. We don’t normally think of spacetime as an entity. But the answers you get when you separate space and time are impossible so something has to give.

I think I got it! Thank you, all!

This explanation of SR tweaks me, because it is kind of true but misses important details and gives a wrong impression because of it.

The norm of the 4-velocity is (or can be defined to be) c, leading to the interpretation of “everything is traveling at the speed of light”, but it is the Minkowski norm that is conserved between reference frames (i.e. ||(a, b, c, d)||[sup]2[/sup] = a[sup]2[/sup]-b[sup]2[/sup]-c[sup]2[/sup]-d[sup]2[/sup]) not the regular Euclidean norm. So people read this explanation and think of a vector that points straight into the future for you, but is simply rotated for another observer the same way one could rotate a vector in space, which is not quite the case.

It’s a standard point of mass (or even massless) confusion, much and often discussed here and everywhere: Earth-bound Terrence and star-bound Stella travel at near-light velocities relative to each other. Which one of them sees the other’s time slowed down, and why? If BOTH of them each see the other’s time slowed down, then which one will be older when Stella returns to Earth?

It’s called the Twin Paradox, and here’s a fairly clear and detailed discussion, which I found from a link right here on SDMB during some earlier discussion of this sort of stuff:

Read it, and be thy ignorance thus fought! (Hey, it worked for me when I read it!)

(Long story short: Terrence, on Earth, is fairly stationary, not accelerating (ignoring details like Earth’s orbital acceleration around the Sun and the whole solar system’s movement through the galaxy), while Stella accelerates leaving Earth, decelerates upon approaching the distant star, and then reverses all that for the return trip. Stella’s variable speeds, with various accelerations/decelerations, versus stationary Terrence, makes all the difference.)

**leahcim**, you’re right that Lorentz transformation isn’t exactly the same thing as Euclidean rotation. But Euclidean rotation is the closest familiar thing we have to it, so saying “it’s sort of like a rotation” is about the best analogy you’re going to find.

**Senegoid**, my preferred way of explaining the resolution to the Twin Paradox is by assuming the astronaut is hitchhiking. Let’s say that Alice stays at home on Earth, but that Bob hitches a ride out to alpha Centauri with Ch’vorthq the alien (who was already going that direction before he got to Earth, and then keeps on going that direction), and then immediately hitches another ride back home with Dax. There are now three different reference frames in the problem: That of Earth, that of Ch’vorthq, and that of Dax. There isn’t a reference frame for Bob, because he changes reference frames: Sometimes he’s in Ch’vorthq’s reference frame, and sometimes he’s in Dax’s reference frame. You can pick any of those three frames to do the calculations (or some other frame entirely, but that’s just making things harder on yourself), but in all of them, Bob will still be young when he again meets up with Alice.

That’s what mean by it being “kind of true”. I have no objection to even dropping the “sort of like” when referring to it as a rotation for a suitable definition of rotation. It’s just the use of the Euclidean analogy without qualification that irks me. It’s not as bad as the “you can’t go faster than light because you get more massive and can’t be accelerated as much” saw, but it’s still a little bit of a cringe.

Plus, it’s a little bit tautological. If you pick any reasonable metric and any reasonable curve, parametrize that curve with respect to the arc length in that metric, and differentiate, you’re going to get a unit tangent vector at every point along the curve. The physics is in why this one metric describes the universe, but once you have the metric the “constant length tangent vector” property follows by process of math.

I’m not getting the massive **cringe factor** in your quoted statement. Please explain why this would be misleading to someone asking “Why not?” to FTL. I have two ideas about qualifying it, but I’d like to know what you think is misleading or missing.

BTW, I think that when one tries to explain certain things to the mathematically challenged (but honestly curious) inquirer, often you have to sacrifice the complete picture for something that you can successfully communicate.

Under any reasonable definition of “mass”, mass does not change for an object moving at high speed.

The origin of this idea is the observation that, at high speed, the formula for momentum is P = gamma*m*v, instead of the familiar P = m*v. Some folks decided that they wanted the formula to look more like the familiar one, and so defined a new quantity called the “relativistic mass”, which is equal to the rest mass times gamma. But there’s no particular reason to attach the gamma to the m, and it leads to all sorts of other misconceptions. If you really want to make the formula look more familiar, it makes much more sense to attach the gamma to the v, so you have P = m*u, where u = gamma*v.

Besides which, we already had a perfectly good term for the quantity gamma*m: That’s just the energy (well, to within a couple factors of c, but those are generally ignored anyway when doing relativity). And meanwhile, the rest mass is a very important quantity, important enough to deserve the nice simple label of “mass”, rather than co-opting that term for a different quantity that already had its own unambiguous term anyway.

I knew **Chronos **would be around quick with the boilerplate relativistic mass post.

Other than those points, two things jump out at me as to why it doesn’t really work as even a rough explanation:

[ol]

[li]We know there are massless particles, and “relativistic mass” is zero for those particles no matter what speed they’re going, so why can’t they go faster than light?[/li][li]The argument is all well and good if someone else is trying to push me faster than light while they’re remaining stationary (like in a particle accelerator or something), but when I’m going at high speeds, I’m doing it in a rocket and my rocket engine is at rest relative to the rocket at all times so it shouldn’t “see” the increased mass. So why can’t that engine accelerate me faster than light?[/li][/ol]

Mostly though, that explanation just avoids all of the meaty parts of SR with the reference frames and the Galilei transformation being incorrect, and turns it into this theory of “fast things get more massive”. It’s not just a simplification, but a pedagogical dead end.

My preferred one-sentence, glossing-over-the-details answer to “why can’t we go faster than light?” is “Because if you go faster than light, then someone in another reference frame would see you going backwards in time, which is impossible”.

Not exactly. Particles with no rest mass also have an infinite gamma factor, so an attempt to define their “relativistic mass” as gamma times their rest mass is mathematically ill-defined. One can define a (finite, nonzero) “relativistic mass” for them as being equal to their energy, but again, if you’re going to do that, why not just call it “energy” to begin with?