I have another question.... What if I'm traveling at 66% of c and someone else is traveling at 66% of c in the oposite direction? I will see him traveling faster than c?
No, if you round both speeds to 2c/3, you'll see him traveling at 12c/13 (in special relativity). More generally, the composition of any two subluminal velocities will always result in a subluminal one.
What happens is that you will see the rest of the universe compressed in the direction you are traveling, and when you measure the speed of the other object, it will be under c.
(This is the reason why you can travel from one point to another in an arbitrarily short period of time from your own perspective, even though from an outsider's perspective your speed never exceeds c. You can travel from Earth to Alpha Centauri in one second, make a 180-degree turn and come back in another second. However, from Earth's perspective your round-trip takes over 8 years. When you come back you will be 2 seconds older, while everyone on Earth will be over 8 years older. Effectively you have traveled to the future. Relativity is wonky like that.)
I have another question.... What if I'm traveling at 66% of c and someone else is traveling at 66% of c in the oposite direction? I will see him traveling faster than c?
let's assume i'm on a spaceship travelling at c and I start to run (about 10 Km/h) someone from the outside will see that I am not moving because the time inside the spaceship will be stopped from the point of view of the person outside the spaceship.
I have another question.... What if I'm traveling at 66% of c and someone else is traveling at 66% of c in the oposite direction? I will see him traveling faster than c?
let's assume i'm on a spaceship travelling at c and I start to run (about 10 Km/h) someone from the outside will see that I am not moving because the time inside the spaceship will be stopped from the point of view of the person outside the spaceship.
Actually I think that the outside observer will also see your spaceship compressed to zero length, which is also why he doesn't see you moving.
Interesting question: If you move at c, does that mean you can travel from any point in the universe to another in zero seconds (from your own point of view)? If the answer is yes, how does this correlate with the so-called cosmological horizon of an expanding universe (which basically means you can't travel from one point to all possible points in the universe, only to another point inside the observable portion of the universe)?
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Of course I don't know how to answer your question warp :)
The problem here is that I asked what would one "see". Is that the problem? That you can't see something traveling faster than c, but something CAN travel faster than c? What I meant was if something can travel faster than c (relative to something else), not if you can see it. Sorry about that.
The problem here is that I asked what would one "see". Is that the problem? That you can't see something traveling faster than c, but something CAN travel faster than c? What I meant was if something can travel faster than c (relative to something else), not if you can see it. Sorry about that.
In special relativity nothing can travel faster than c. No matter who measures what, nothing travels faster than c. (One way of stating this is that if you send a photon towards something, and leave afterwards to "chase" it, you can never outrun that photon. You can never reach the destination faster than that photon will. And photons will always travel at c in vacuum.)
In general relativity things get a bit more complicated because space is not cartesian and the geometry of space can actually change. The distance between two points can increase faster than c, but this still doesn't break the principles of special relativity because nothing can travel between those two points faster than c.
@FODA: In relativity (special or general), nothing massive can travel at or above the speed of light. In theory, if you have an imaginary mass, you can travel above the speed of light; these are the theoretical tachyons -- which physicists don't like much because they make spacetime unstable. Only things with zero mass (such as the photon) can travel at the speed of light. Moreover, relativity also prevents stuff from moving from one "category" to another without undergoing changes in mass and composition -- thus, no matter how strong your propulsion system is, or how long it can be kept on, you can't more at the speed of light unless you throw away all your mass, and you can't go above the speed of light unless you acquire some imaginary mass (and the converse if you started out as a tachyon).
The main difference between special and general relativity is that in the latter, you can't move faster than light locally -- that is, you can't outrun that beam of light you just fired -- but you potentially can globally -- you send a beam of light one route, take a different route and still arrive at that distant star faster than your beam of light because the specifics of spacetime curvature gave you a shorter route.
@Warp: at the speed of light, your proper time doesn't "tick". Hence, you can travel to any point in the Universe to any other point in zero time -- as long as these points are separated by a "null" or "lightlike" interval (basically, the spacetime distance between them comes out "zero").
Curved spacetime has the problem that the (spacetime) distance between any two points can have multiple values (and no, taking the minimum doesn't work -- even in the case of the flat spacetime of special relativity -- due to the hyperbolic nature of spacetime).
In theory, if you have an imaginary mass, you can travel above the speed of light
Has any imaginary quantity been observed in nature yet? As I understand it, imaginary numbers are only a math trick needed to solve some equations, not a property of the physics systems they describe. Am I wrong here?
Joined: 5/1/2004
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I think I can't see a relation to what you're saying with what I asked, or you guys are still missing my point. :)
A spaceship is moving at 200 thousand meters per second east, while another object is moving 200 thousand meters per second west. To the crew in the ship the other object is moving at 400 thousand meters per second, faster than light, even though they can't see it. No?
No.
It's been a long time since I did special relativity, but it's filled with these kinds of word problems. The short version is that at relativistic speeds, things get noticeably dilated and contracted -- the rate at which time passes, the size of the object that is moving, the amount of mass it has, etc. all depend on how fast it's going (in fact this is true at any speed, but the factor is negligible unless your speed is a significant fraction of c). And the math works out so that two objects on a head-on collision, each travelling at .99c from a "stationary" perspective, will each appear from the other's perspective to be traveling at .9999c or the like.
I know this is unsatisfying, so with any luck someone who remembers relativity better will give a more thorough answer.
Pyrel - an open-source rewrite of the Angband roguelike game in Python.
I will also add that there are two distinct concepts involved in your question that usually get mashed together because of our Galilean intuitions and cause needless confusion. They are:
Rate of separation between two objects as seen by a third object;
rate of separation between two objects as seen by one of them ("relative speed", "relative velocity").
Our intuition usually lumps those two concepts together -- in our day to day experience, there is no distinction between these two concepts, and (1) and (2) are numerically equal.
But this happens only because the speeds involved in our daily lives are an insignificant fraction of the speed of light in vacuum; when speeds increase, this view leads to errors that grow very quickly.
In relativity, concepts (1) and (2) are related, but not equal: the key aspects are time dilation and spatial contraction. Since time and space are different for each object, (1) and (2) can't be compared in the way our intuition wants to compare them. Both are rate of changes of position over time, but they use "different" rulers (spatial contraction) and "different" scales for time (time dilation).
(2) can never exceed c; spatial contraction and time dilation see to that. This isn't a huge coincidence or conspiracy; merely a mathematical property of any spacetime where you have such a limiting speed. An analogy here is with a rotation -- rotations keep lengths intact, just as changing speeds keep speeds below c (to be honest, this isn't an analogy -- a change of speed, or "boost", is a rotation on a plane that contains the time axis).
On the other hand, there is a much weaker limit to what value (1) can take -- it can range from -2c to +2c (with the extreme cases being light), since each object can't be moving away faster than c from the third object.
Most people encountering relativity for the first time (which includes almost all physics students) have enormous difficulty because the difference between these concepts is not adequately explained -- mostly because they usually aren't explained at all.
An analogy here is with a rotation -- rotations keep lengths intact, just as changing speeds keep speeds below c (to be honest, this isn't an analogy -- a change of speed, or "boost", is a rotation on a plane that contains the time axis).
Oh man, I'd forgotten about that explanation. Yeah, it's a great "intuitive" way to think about relativistic motion. You're moving at a constant rate through 4D space-time; the rate at which time passes (your speed along the time axis) is greatest when you're sitting still, but as you increase your speed in 3D space, your speed in time decreases. Easier to think about with 2D space and time, since you can imagine your velocity as being on the position of a hemisphere. When X and Y velocities are 0, T velocity is 1; when sqrt(X^2 + y^2) is 1, then T velocity is 0 (you're traveling at the speed of light). Because of the way square roots work, you can get going pretty quickly from a classical perspective along X and Y before T starts to fall off noticeably (i.e. the hemisphere is locally flat).
Pyrel - an open-source rewrite of the Angband roguelike game in Python.
I think I can't see a relation to what you're saying with what I asked, or you guys are still missing my point. :)
A spaceship is moving at 200 thousand meters per second east, while another object is moving 200 thousand meters per second west. To the crew in the ship the other object is moving at 400 thousand meters per second, faster than light, even though they can't see it. No?
Your question was understood and answered the first time you asked it. Please try to understand the answers.
Granted, relativity is not intuitive because it does not conform to everyday experience. The world just starts acting "weird" when something starts moving at very great speeds (much greater than what we are accustomed to in everyday life). And this is not just a hypothesis, it's a measured fact.
To be fair, his question was only answered in this post. Indeed the two objects would be separated at 400 km/s while each one is moving at 200 km/s, regardless of observers (which are irrelevant as per his premise).
Warp wrote:
Edit: I think I understand now: It's my avatar, isn't it? It makes me look angry.
To be fair, his question was only answered in this post. Indeed the two objects would be separated at 400 km/s while each one is moving at 200 km/s, regardless of observers (which are irrelevant as per his premise).
Not quite; specifying the observers is of paramount importance with anything having to do with speed in relativity -- except for the speed of light, all speeds are relative, and it is meaningless to speak of speed without specifying an observer (technically, this was true of Galilean relativity too, but the distinct concepts I mentioned in my other post are equal there, and the "invariant speed" is infinite). For an observer that sees both objects moving at 200 km/s toward each other, yes, they will be approaching each other at 400 km/s; for any other observer, the speed of approach will be different.
Lets for simplicity take speeds as fractions of the speed of light; then, 200 km/s is equivalent to 66.67%. Using nothing but the relativistic formula for velocity composition:
u' (u - v) / c
--- = -------------
c 1 - uv/c^2
(here in a form suited to using speeds as % of c) we can obtain the following table:
Here, I assume that an "observer speed" of 0% corresponds to an observer that sees both objects moving at a speed of 66.67% (line marked *); each line corresponds to another observer moving relative to that reference observer.
"Object # speed" refers to the speed of object # relative to the current line's observer. This would fall in concept (2) in my other post. These two columns are always less than c.
"Approach speed" is how fast object 1 approaches object 2 according to the current line's observer. This would fall in concept (1) in my other post. This is what I said is usually conflated with relative speed; it is "uncapped" except indirectly (because the object speeds are capped). As can be seen, it varies greatly, and depends heavily on the observer.
"Relative speed" is the speed with which object 1 sees object 2 moving. This would fall in concept (2) in my other post. This column is what p4wn3r was referring to in his post; this was also what FODA was asking about. This one is constant -- but that is not because I made all entries equal, because I didn't! I put a formula in this column just as I did in all but the first column; the fact that they all come out equal is just a reflection of the self-consistency of the theory. See below for details on how I built the table.
The lines marked with ** are the rest frames of either one of the moving objects; note that approach speed and relative speed become equal in these cases.
As you can see, the objects can't be said to be approaching at a speed of 133.33% (400 km/s) except for one observer; and for sufficiently fast-moving observers (relative to line *), they are approaching much slower than that (the extreme lines, for example). This table also ignores motion on any direction other than the line between the two objects; were such motion considered, the table would be very, very different.
As for the table: it was done in LibreOffice Calc, with the "observer speed" column in column 1 (first number at A1) using the following formulas in the other columns:
B2: =(B$13-$A2)/(1-B$13*$A2)
C2: =(C$13-$A2)/(1-C$13*$A2)
D2: =B2-C2
E2: =(B2-C2)/(1-B2*C2)
then drag/copy to all rows except row 13. Column A is for "initial values", as are B13 and C13; everything else used the formulas above. I configured all rows to show percentages, then hand-formatted the table after pasting it in a text editor.
