
If two objects approach at 75 percent the speed of light - mavdi
http://einstein.stanford.edu/content/relativity/q1021.html
======
Steuard
This explanation is correct, but I find it pretty empty of intuition: "Here's
a magic formula! Look, it adds up funny." I've taught relativity quite a bit
recently, and I much prefer to build intuition using "space-time diagrams" (or
"Minkowski diagrams") showing coordinate systems for two different observers.

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](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.

~~~
Retric
IMO, a much more natural approach is to look at things from the perspective of
light emitted. Specifically black body radiation. Red/Blue shifting correlates
with changing temperatures. And temperature works as a natural proxy for time.

The other thing that confuses people is presenting things from an
'independent' perspective in the diagram.

~~~
figure8
If I'm not mistaken, the derivation of the black-body radiation energy
distribution does not use special relativity at all, only statistical
mechanics and quantum mechanics. How can it be used to explain special
relativity?

------
hliyan
I find this explanation pretty intuitive:

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).

~~~
yAak
Yay, your explanation makes perfect sense to me! This part: "...relative to
Earth, my brain and my instruments are running slower due to time dilation"
helped a lot with clarity and now it's intuitive for me as well.

The linked page just managed to confuse me, so thanks for posting this!

------
amirmc
> _" Some problems that we all have in understanding how nature works, have
> more to do with our conviction that our intuitions are correct, despite the
> fact that our intuitions are often flawed in areas of experience outside of
> our normal environment."_

This really stood out to me. It's applicable to more than just the current
example.

~~~
zak_mc_kracken
True. What makes this model compelling is that our flawed intuitions are very
good at describing the world we live in, which is macroscopic and non
relativistic.

Letting go of these intuitions is a requirement to become a competent
physicist.

~~~
Retra
You replace them with new intuitions about what kinds of transformations are
allowed. (Like unitarity.)

------
alberte
The question should be "If two objects approach at 75 percent the speed of
light _relative to me_ then what is their speed relative to each other"

~~~
EricSu
I agree...I know this is supposed to be about their relativity to each other
but the way my mind interpreted the wording was "Why are neither of the two
objects traveling at 1.5 times the speed of light?" and I immediately thought
"Well because it's relative and you can't say that you driving 60mph and
another car going 60 mph in the opposite direction means you're driving at
120mph because you're still driving 60mph

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.

~~~
tcgv
If my car is travelling at 60 mph and there is a car in the opposite direction
also travelling at 60 mph (according to a inertial frame of reference) then
classical mechanics allows me to say that from my point of view the other car
is indeed traveling at 120 mph. But special relativity only allows me to say
that the other car is travelling at 119.9999999999990394 mph from my point of
view. The higher the speed, the greater the difference in the results from
"classical mechanics" and "special relativity" will be.

------
tlogan
The explanation really does not help people who are not familiar with this
topic.

I find that the following Susskind's lectures explain this subject really
well:
[https://www.youtube.com/playlist?list=PLDDFE71BA2DE55505](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?

~~~
octatoan
That's a nice way to put it!

------
peter303
Unsatisfactory answer, though mathematically correct. The fuller explanation
considers what constitutes "measurement" in a physics with a speed limit
(lightspeed). We measure "intervals", i.e. lengths between to points of space
or time. We'll see the intervals stetched/contracted by the Lorentz factor in
another frame moving at a high constant velocity.

------
learnstats2
So, if two objects approach each other at 60mph, they are actually travelling
at 119.9994mph relative to each other?

~~~
mavdi
No, much closer to 120.

120 / (1 + (60 / 670616629) * (60 / 670616629)) ~= 119.9999999999990394

~~~
snowwrestler
The difference between this speed and 120 mph is less than half a picometer
per a second, which illustrates why we don't typically notice relativistic
effects at highway speeds. :-)

------
skierscott
Nothing can exceed the speed of light. This equation represents that.

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).

~~~
spacehome
Causality is always preserved. If A happens in the past light cone of B in one
frame, then it happens in the past light cone of B in all frames.

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.

------
amelius
Here is a phenomenon that is truly fascinating:
[https://www.youtube.com/watch?v=wteiuxyqtoM](https://www.youtube.com/watch?v=wteiuxyqtoM)

------
dm2
Here are several related images and graphs:

[https://www.google.com/search?q=relative+velocity+time+dilat...](https://www.google.com/search?q=relative+velocity+time+dilation&source=lnms&tbm=isch)

[https://www.google.com/search?q=velocity+special+relativity&...](https://www.google.com/search?q=velocity+special+relativity&source=lnms&tbm=isch)

[http://hyperphysics.phy-
astr.gsu.edu/hbase/relativ/einvel.ht...](http://hyperphysics.phy-
astr.gsu.edu/hbase/relativ/einvel.html)

------
strags
I found this extraordinarily helpful:

[http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf](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.

------
btilly
Useless trivia.

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](http://en.wikipedia.org/wiki/Fizeau_experiment)
for details.

------
z3t4
If there actually where something moving faster then light, and you measured
the speed, it would still appear to be moving at the speed of light!

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?

~~~
sqeaky
Depends on what you mean by FTL. Most Scifi canons take FTL to mean "ignores
relativity", like Star Trek and Star Wars.

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.

------
astrobe_
So if those objects are on a collision course, they will crash on each other
at 0.96c instead of 1.5c?

~~~
marcosdumay
Yes, but not problem, they'll release all the energy they've used to
accelerate so the collision will be exactly as amazing.

------
ck2
What really blows my mind about it all is we didn't even have the math to
figure that out until 100 years ago (Einstein 1905)

------
return0
Don't forget this year is the centennial of the General theory of Relativity

------
s_kilk
You will be removed from the universe, for violating it's laws.

~~~
coldpie
Hello, friend! "It's" is a contraction, usually for "it is" or "it has." "Its"
is the possessive form of "it," just like "hers" and "his." "Its" is used to
show that something is owned by "it." In this case, the laws belong to the
universe, so you want the possessive form: "for violating its laws."

Thanks for reading!

~~~
moron4hire
Sometimes my phone autocorrects it incorrectly on me, and I haven't noticed
before I tap the "reply" button.

~~~
shogun21
Autocorrect doesn't know sometimes "its" is the correct form.

~~~
oaktowner
Off-topic, but: yes! One of my big annoyances with autocorrect is it auto-
uncorrecting "its" to "it's", "well" to "we'll", etc. Not sure what algorithm
is used, but it seems to be fairly horrible (BTW - I'm on a Moto X now and
used to have an iPhone; I've seen it in both).

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.

