If two objects approach at 75 percent the speed of light
einstein.stanford.edu
einstein.stanford.edu
Tom Moore's Six Ideas That Shaped Physics textbook (Unit R) does a very nice job of this. But here's a really bargain-basement handout that I made many years ago to explain the idea before I started using Moore's text for this. (I'd probably do it a little differently now.) http://www.slimy.com/~steuard/teaching/classes/spacetime.pdf
My discussion of velocity addition there (at the bottom of page three out of four in that PDF) is much too terse for this context, I'm afraid. But if you draw my example there carefully and measure in the tilted coordinates as described earlier, you'll start to see why the leftmost observer still measures the rightmost observer moving less than the speed of light.
The other thing that confuses people is presenting things from an 'independent' perspective in the diagram.
At what speed does the point of contact move, relative to the reference frame of either bar, or to a static external observer?
In that case, the "static observer" would see the point of contact moving at very nearly sqrt(2) times the speed of light (along the x-y diagonal). But that's okay, because no physical object (or bit of information) is being carried along at that intersection point. (If you put some little spindle at the intersection to be carried along, you'd suddenly have Problems.)
On the other hand, from the reference frame of one moving bar, the intersection point appears to be moving almost infinitely fast! If the "static observer" measured speeds u and v for the two rods (as fractions of light speed, so u and v are both close to 1), then an observer moving with one rod will see the intersection point move at speed v/sqrt(1-u^2). If we write u = 1 - ε for some very small ε, then the intersection point's speed in this frame is roughly v/sqrt(2ε). (The intuitive reason for this result is interesting: an observer on one moving rod will consider the other rod to be almost parallel, at a very small angle (rather than perpendicular), and still rushing toward her at nearly the speed of light.)
What's your field?
Then the answer is just the speed of light in both cases, and that's really the crux of the problem of special relativity and that the speed of light is constant in all references frames. Then you work backwards to see that the observer that is moving must have time dilation effects.
I also liked the geometrical approach to Minkowski space-time a lot better than just chewing through formulas -- like Taylor and Wheeler's Space-Time Physics book. Drawing light cones and various grids and curves of constant space or time measurements relative to different observers made it all a lot more concrete to me.
1. I left Earth at 0.75c and haven't accelerated or decelerated since
2. I receive a message from Earth that an object is heading my way at 0.75c (far enough away for them to message me at 1c before it hits me)
3. I look where they point and see the object approaching at 0.96c
4. I'm not confused because I fully realize that relative to Earth, my brain and my instruments are running slower due to time dilation. Speed is a time-related measure (d/t) -- you can't use relative distance and absolute time to calculate (d/t). You use relative values for both. So the speed I get when I divide by my t is different to what Earth gets when it divides by it's t.
The problem arises when people accidentally assume an absolute vantage point to observe the two objects. A scientist on Earth will see the gap between the 2 objects close at 1.5c, but he'll know better than to make the usual assumption we make on Earth -- that everyone else will see the gap close at the same rate (because time will be slower for some observers).
The linked page just managed to confuse me, so thanks for posting this!
This really stood out to me. It's applicable to more than just the current example.
Letting go of these intuitions is a requirement to become a competent physicist.
edit: Because the question doesn't specify the speed relative to what exactly, I automatically thought about the speeds of the objects relative to themselves rather than to each other or a 3rd party.
I find that the following Susskind's lectures explain this subject really well: https://www.youtube.com/playlist?list=PLDDFE71BA2DE55505
Basically, the key of understanding of special relativity is to start with: Maxwell's equations are correct. But these equations also need to be correct in every reference frame - even if speed of light is constant. How to fix that?
120 / (1 + (60 / 670616629) * (60 / 670616629)) ~= 119.9999999999990394
I also seem to remember that causality is not preserved. This means that if A happens before B in one frame, it's possibly that B happens before A in another frame.
I also seem to remember that if it's possible to break the speed of light, it'd be possible for B to fire before A (if A triggers B, all in one frame).
But causality itself is safe in relativity: if a signal from event A could possibly reach event B (given the speed of light limit), then every reference frame will agree that A comes before B. The only events whose relative order is indeterminate are those separated by enough distance (relative to elapsed time that one couldn't influence the other: events who are not in each other's "light cones".
If, on the other hand, neither A nor B are in each other's past light cones, then there are reference frames where A or B happen 'first', but 'first' is in quotes, because this difference in time is really an artifact of a coordinate choice, and not physically meaningful.
The best way to think about time relationships is that events (points in spacetime) have a partial order given by the relationship 'is in the past lightcone of'. Some pairs of points have this relationship one way or the other, and it's transitive. However some pairs of points don't have this relationship, and there's no meaningful way to compare them. Attempting to do so is inappropriately forcing a classical viewpoint onto a data model that doesn't support it.
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf
The Lorentz transform allows you to calculate exactly how time/distance differ between two reference frames.
The above link derives the transform from first principles - assuming only spatial symmetry and the invariance of the speed of light in different reference frames. There are other derivations, but I found this one the most straightforward.
Did you know that fact that speeds do not simply add together at relativistic velocities was directly measured in 1851? The result of that measurement was the belief that there was some sort of "partial frame dragging" of the ether. The exact details of which got more and more confusing as experimenters looked farther into it.
See http://en.wikipedia.org/wiki/Fizeau_experiment for details.
Say you had two space ships with FTL capability that would race from our solar system to the finish line located near Alpha Centauri. How would you know witch one came first?
In this context someone using a relativistic viewpoint would simply see the ship disappear, then appear somewhere else before they left (using a huge telescope or whatever and calculating the time difference).
The results of return trip are totally based on the interactions of FTL with relativity as we know it. It could be totally ridiculous.
Imagine the a ship here leaves for Alpha Centauri (AC) at hyperspeed or warp 9. For this discussion lets assume that the travel takes 10 minutes. But for light it would take 5 years (rounding).
This functionally would be travel 5 years into the past. So earthbound observatories would see the ship right now, because it left the Falcon/Enterprise 5 years ago from a relativistic stance. Then the ship could return and see its own light given off at alpha centauri?
What does this look like from the viewpoint of observatories on Alpha Centauri(AC)?
Thinking about it, it would have to look the same. By definition FTL is faster than light, but light is attached to time. So the trip from Sol to AC must either go back 5 years or otherwise reconcile our 5 year clock skew.
I don't see how any reconciliation mechanic makes any sense.
But yeah you are right that a ship that simply accelerates so much that its riders think and measure their own speed as faster than light are observed by outsiders as hugging light speed.
Thanks for reading!
The only way to do this would be to wait until enough of the sentence has been written to get context. Failing that, though...I just don't want it to touch those words at all.