Two spaceships in opposite direction at near c

[stu dent]

if you have 2 spaceships and they depart in exactly the opposite direction from a starting point, say a space station, and they accelerate to speeds nearing the speed of light, then what is the relative speed of each spaceship, using one of them as a reference frame.

it would seem to me, that the space station would be moving at a speed nearing c, and then the other spaceship would need to be traveling at double that.

but that can't be right.

 

[PeterDonis]

if you have 2 spaceships and they depart in exactly the opposite direction from a starting point, say a space station, and they accelerate to speeds nearing the speed of light, then what is the relative speed of each spaceship, using one of them as a reference frame.

it would seem to me, that the space station would be moving at a speed nearing c, and then the other spaceship would need to be traveling at double that.

but that can't be right.

Yes, it can't be right, and it isn't. The correct formula for adding velocities in SR is:

$$w' = \frac{v + w}{1 + vw / c^2}$$

where w is some object's velocity in one frame, and w' is the same object's velocity in a second frame moving at v relative to the first.

In your scenario, say spaceship A moves off to the left from the space station, with speed w, and spaceship B moves off to the right with speed v. In spaceship B's frame, the velocity of spaceship A is then w', as given by the above formula. It should be evident that if w < c and v < c, then w' < c also.

Picking some concrete numbers for an example, if v = .99c (to the right) and w = .99c (to the left), then

$$w' = \frac{.99 + .99}{1 + .99 * .99} c = .99995 c$$

So spaceship A will be moving at .99995c relative to spaceship B.

 

[stu dent]

Yes, it can't be right, and it isn't. The correct formula for adding velocities in SR is:

$$w' = \frac{v + w}{1 + vw / c^2}$$

where w is some object's velocity in one frame, and w' is the same object's velocity in a second frame moving at v relative to the first.

In your scenario, say spaceship A moves off to the left from the space station, with speed w, and spaceship B moves off to the right with speed v. In spaceship B's frame, the velocity of spaceship A is then w', as given by the above formula. It should be evident that if w < c and v < c, then w' < c also.

Picking some concrete numbers for an example, if v = .99c (to the right) and w = .99c (to the left), then

$$w' = \frac{.99 + .99}{1 + .99 * .99} c = .99995 c$$

So spaceship A will be moving at .99995c relative to spaceship B.

i see. thx, but by what mechanism does this relationship exist? meaning, what is it in the real world is this formula explaining, or, where does it come from?

 

[arindamsinha]

i see. thx, but by what mechanism does this relationship exist? meaning, what is it in the real world is this formula explaining, or, where does it come from?

Special Theory of Relativity.

 

[PeterDonis]

i see. thx, but by what mechanism does this relationship exist? meaning, what is it in the real world is this formula explaining, or, where does it come from?

The formula I gave is the relativistic velocity addition law, which is a consequence of the Lorentz transformation formulas. See the Usenet Physics FAQ for a good (if brief) discussion:

http://math.ucr.edu/home/baez/physics/Relativity/SR/velocity.html

 

[bobc2]

i see. thx, but by what mechanism does this relationship exist? meaning, what is it in the real world is this formula explaining, or, where does it come from?

stu dent, one way to get a physical feel for what's going on is to develop an understanding of your problem in the context of a 4-dimensional universe. I'll give you a quick picture and a link for further details just in the event you might be interested in this approach. I and others here can provide more information on this if you wish. One key element of the picture below is a strange and mysterious aspect of nature resulting in different cross-section views of the universe for observers moving at different speeds. This, in large measure, accounts for the strange things going on with special relativity theory and why velocities do not add the way you might have thought. The slant of the X4 axes for the red and blue rockets relate to speeds--but we have red, blue and black frames of reference here, and the X1 axes (worlds the different observers "live in") cut across the universe at different angles as well.

You might begin by looking at the picture illustrating the different times for the red and blue observers (time dilation) as shown below. When the blue guy is at position 9 along his X4 axis (time axis) his world includes the red rocket located at red's frame position 8. But when red is at his position 9, the blue rocket is at the blue position 7. Keep in mind each is viewing 3-D cross-sections of 4-dimensional rockets.

Here is a link for more details (see post #19):
https://www.physicsforums.com/showthread.php?p=4138850#post4138850

 

[bobc2]

If you were able to follow the space-time diagrams in the previous post and link, then you can easily see in the sketch below how it is that the velocities add. In the previous post we had red and blue frames moving in opposite directions at the same speed with respect to the black frame. Now we look at the speeds of the red and black frames with respect to the blue frame. The black frame moves at 1/2 the speed of light with respect to the blue frame (moving in blue's negative X1 direction). The red frame is moving at 1/2 the speed of light with respect to the black frame (moving in black's negative X1 direction).

But, now you see clearly that (1/2)c + (1/2)c does not give you 1 x c for the red frame with respect to the blue frame. Red is moving at relativistic speed in the blue negative X1 direction, but certainly not at the speed of light.

 

[harrylin]

Special Theory of Relativity.

In addition: SR does not give a "mechanism", as it is based on logical (mathematical) deduction of phenomenological principles. Historically there have been different explanations of "what is really going on". If Bob's 4D reality makes sense to you, then that is nice; alternatively you could go for Lorentz's "ether" concept or Harvey's "physical relativity" or ...

 

[stu dent]

Special Theory of Relativity.

no offense, but you may as well have said: because i said so. Which will never be satisfactory for me, even if it is einstein that said so.

i mean, not that i doubt einstein is correct, but i want to know why he must be correct. why it is necessary for it to be that way.

i mean, i realize it must be many steps of logic building upon each other in order to arrive at this conclusion, but there are steps that provide certainty and conclude this fact.

if i am in a space ship, and i travel in one direction at nearly the speed of light, and another does so in the opposite direction, then i don't see how i influence that other space ship. i don't understand how my velocity, can hold implications for the velocity of another object. it makes perfect sense to me that an energy with mass cannot reach the speed of light. i get that.

but i don't fully get why or how it can be that my velocity relative to another object that has mass can't reach or exceed the speed of light.

what I'm thinkign now, that i will ponder further also, is that it nearly seems as though the quantity of energy required to accelerate an object would then be different depending on your reference frame.

or, since this seems not to make a lick of sense, the relative velocity of (let's call it spaceship A and B and then Earth) of A relative to earth, is the equivalent to A relative to B.

or no, wait, A relative to light in Earth frame, is equivalent to A relative to light in B frame, which may or may not be different, which i also need to think about, and also, a curious thing would seem to me, that the closer that B approaches the speed of light then, regardless of the velocity of A, the more similar A's velocity is to Earth's in B's reference frame.

which intuitively feels to me, like the opposite of time dilation effect, because it is like putting the universe on pause, while you continue to age.

 

[stu dent]

thanks for all the replies, i will explore these links and posts, but may not get to reply until later.

 

[PeterDonis]

if i am in a space ship, and i travel in one direction at nearly the speed of light, and another does so in the opposite direction, then i don't see how i influence that other space ship. i don't understand how my velocity, can hold implications for the velocity of another object.

It doesn't. That's not the issue. The issue is, how do you *measure* the velocity of an object that is spatially distant from you? You can't observe it directly, so you have to somehow translate the direct observations you can make into a value for the distant object's velocity. Any such translation is not just dependent on your motion, or on the motion of the distant object; it's also dependent on the properties of spacetime itself, since the information about the distant object's motion has to travel to you through spacetime. And it's the properties of spacetime that give rise to the relativistic velocity addition formula.

 

[stu dent]

In addition: SR does not give a "mechanism", as it is based on logical (mathematical) deduction of phenomenological principles. Historically there have been different explanations of "what is really going on". If Bob's 4D reality makes sense to you, then that is nice; alternatively you could go for Lorentz's "ether" concept or Harvey's "physical relativity" or ...

i think maybe mechanism was not the best word, what i was trying to convey was a hard thing to say. i meant like, why it must be.

math is all very nice. but math is not enough. it is not a proper explanation.

for example, we could plot 4 dimensional equations in a 2d altitude map kind of way, but for 4d, and we'd get some confusing drawings. the understanding would be lost on us. we could do operations as well, and mathematical deduce the results of these things, and try to draw more kind of maps again as well.

but still, we would know nothing.

but call the 4th dimension time, and suddenly everything changes. the realisation, which is not mathematical, makes everything apparent. it allows it to make sense. it provides a real world understanding. we were plotting a moving object in time frames.

i was getting at something like that. i want to understand how it can be possible, and why it must be possible, and what exactly is happening.

the formula should be telling for this, but i am uncertain how it got discovered, why it must be that way.

the formula is precise directions of how. it does not show why, nor does it show why it is certain that is how.

 

[stu dent]

how did lorentz figure out lorentz transformations, and why they were necessary?

 

[Warp]

how did lorentz figure out lorentz transformations, and why they were necessary?

The lorentz transformation can actually be derived by simply starting with the assumption that the speed of light in vacuum is the same for all observers (which, in fact, is an observed and measured phenomenon, rather than being a mere hypothesis). It isn't even all that complicated to understand how it's derived from there.

The factor that Lorentz came up with was simply a description of how relative lengths contract in different frames of reference (and, as said, is derived directly from assuming that c is the same in all frames of reference.) It described the results of the Michelson-Morley experiment as well as a bunch of others.

(This was technically speaking a hypothesis at first, because there was only a limited amount of measurement data available. However, with time it has been proven quite accurate in quite many different situations.)

 

[harrylin]

how did lorentz figure out lorentz transformations, and why they were necessary?

He was developing an electrodynamics theory (electrons etc) which modeled electrodynamics relative to a stationary medium, the "ether". Following the "Galilean" transformations of classical mechanics, it should be possible in theory to detect motion relative to the ether - but such experiments failed.

In earlier discussions I gave a much simplified sketch of how he figured it out:
https://www.physicsforums.com/showthread.php?p=3756233#post3756233 (very roughly the same approach as Einstein)
https://www.physicsforums.com/showthread.php?p=3887942#post3887942 [/url] (simplified and incomplete derivation; and see the correction in the there following post).

 

[robinpike]

no offense, but you may as well have said: because i said so. Which will never be satisfactory for me, even if it is einstein that said so.

i mean, not that i doubt einstein is correct, but i want to know why he must be correct. why it is necessary for it to be that way.

i mean, i realize it must be many steps of logic building upon each other in order to arrive at this conclusion, but there are steps that provide certainty and conclude this fact.

if i am in a space ship, and i travel in one direction at nearly the speed of light, and another does so in the opposite direction, then i don't see how i influence that other space ship. i don't understand how my velocity, can hold implications for the velocity of another object. it makes perfect sense to me that an energy with mass cannot reach the speed of light. i get that.

but i don't fully get why or how it can be that my velocity relative to another object that has mass can't reach or exceed the speed of light.

Student, don't give up on seeking the answer to that question - that question shows great insight.

 

[bobc2]

Student, don't give up on seeking the answer to that question - that question shows great insight.

A wonderful and refreshing response, robinpike.

 

[robinpike]

if i am in a space ship, and i travel in one direction at nearly the speed of light, and another does so in the opposite direction, then i don't see how i influence that other space ship. i don't understand how my velocity, can hold implications for the velocity of another object. it makes perfect sense to me that an energy with mass cannot reach the speed of light. i get that.

but i don't fully get why or how it can be that my velocity relative to another object that has mass can't reach or exceed the speed of light.

Here is an example of the issue:

Two space ships accelerate in opposite directions from a space station in deep space. The space station calculates the speed of each departing spaceship relative to itself by sending out light signals to the space ships, which they return straight back to the space station.

In this way, the people on the space station can calculate the speed of each spaceship relative to themselves. Eventually, the two space ships reach individual departing speeds from the space station of 3/4 the speed of light.

This suggests that the people on the space station see the two space ships recede away from each other at 1.5 times the speed of light?

And yet the two space ships can still communicate with each other by sending light signals to each other.

 

[harrylin]

Here is an example of the issue:

Two space ships accelerate in opposite directions from a space station in deep space. The space station calculates the speed of each departing spaceship relative to itself by sending out light signals to the space ships, which they return straight back to the space station.

In this way, the people on the space station can calculate the speed of each spaceship relative to themselves. Eventually, the two space ships reach individual departing speeds from the space station of 3/4 the speed of light.

This suggests that the people on the space station see the two space ships recede away from each other at 1.5 times the speed of light?

And yet the two space ships can still communicate with each other by sending light signals to each other.

That is not an issue (and different from the OP), except if it is still an issue for you if your replace "light in space" by "sound in air". There is in principle no problem for two airplanes to exchange sound signals if they recede away in opposite directions at 3/4 the speed of sound.

 

[ghwellsjr]

Yes, that's because, as PeterDonis pointed out in post #2, they see each other receding away from each other at:

w' = (.75 + .75)/(1 + .75 * .75) c = .96 c

 

[jartsa]

Here is an example of the issue:

Two space ships accelerate in opposite directions from a space station in deep space. The space station calculates the speed of each departing spaceship relative to itself by sending out light signals to the space ships, which they return straight back to the space station.

In this way, the people on the space station can calculate the speed of each spaceship relative to themselves. Eventually, the two space ships reach individual departing speeds from the space station of 3/4 the speed of light.

This suggests that the people on the space station see the two space ships recede away from each other at 1.5 times the speed of light?

And yet the two space ships can still communicate with each other by sending light signals to each other.

Let us ask the question: How much can a photon's velocity change?

-300000 km/s - 300000 km/s = -600000 km/s

A photon approaching a spaceship, that is moving to the left very fast, has velocity -300000 km/s

After this photon has reflected from the spaceship it has velocity 300000 km/s

Before the reflection the photon and the spaceship had about the same velocity, then the velocity of the photon changed by 600000 km/s.

 

[bobc2]

Let us ask the question: How much can a photon's velocity change?

-300000 km/s - 300000 km/s = -600000 km/s

A photon approaching a spaceship, that is moving to the left very fast, has velocity -300000 km/s

After this photon has reflected from the spaceship it has velocity 300000 km/s

Before the reflection the photon and the spaceship had about the same velocity, then the velocity of the photon changed by 600000 km/s.

Nice idea, jartsa. But, are you sure it was the same photon?

 

[jartsa]

Nice idea, jartsa. But, are you sure it was the same photon?

Let me rephrase the idea, not using the word photon:

300000 km/s is a very important constant in physics.

600000 km/s is a somewhat useful constant:
Light sphere's diameter can increase by 600000 km/s.
Light beam's velocity can change by 600000 km/s.

 

[ghwellsjr]

Let me rephrase the idea, not using the word photon:

300000 km/s is a very important constant in physics.

600000 km/s is a somewhat useful constant:
Light sphere's diameter can increase by 600000 km/s.
Light beam's velocity can change by 600000 km/s.

I'm not sure what you're getting at here but if you're following Special Relativity, light always travels at 300000 km/s in any frame you choose. It's velocity never changes and nothing can travel faster than 300000 km/s.

 

[Drakkith]

Let us ask the question: How much can a photon's velocity change?

-300000 km/s - 300000 km/s = -600000 km/s

A photon approaching a spaceship, that is moving to the left very fast, has velocity -300000 km/s

After this photon has reflected from the spaceship it has velocity 300000 km/s

Before the reflection the photon and the spaceship had about the same velocity, then the velocity of the photon changed by 600000 km/s.

Yes, but so what? It's speed is always the same at 300,000 km/s.

 

[jartsa]

I'm not sure what you're getting at here but if you're following Special Relativity, light always travels at 300000 km/s in any frame you choose. It's velocity never changes and nothing can travel faster than 300000 km/s.

Light beam going to the left has velocity -300000 km/s.
Light beam going to the right has velocity +300000 km/s.

Is this not the normal way of talking about velocities of things?

Light beam going to the left may turn into light beam going to the right, when reflecting, right?

 

[jartsa]

Yes, but so what? It's speed is always the same at 300,000 km/s.

Well, the 600000 km/s velocity change might help some newbies to see how light can travel between two spaceships whose distance is increasing by nearly 600000 km/s. (according to an observer that sees both spaceships moving at near c into opposite directions)

 

[Nugatory]

Well, the 600000 km/s velocity change might help some newbies to see how light can travel between two spaceships whose distance is increasing by nearly 600000 km/s. (according to an observer that sees both spaceships moving at near c into opposite directions)

That's a good point... And now that I think about it, you don't need the reflection off of one the spaceships to make it, just consider two flashes of light moving past each other in opposite directions.

 

[ghwellsjr]

Well, the 600000 km/s velocity change might help some newbies to see how light can travel between two spaceships whose distance is increasing by nearly 600000 km/s. (according to an observer that sees both spaceships moving at near c into opposite directions)

But that would be the wrong way to explain it to a newby.

In an Inertial Reference Frame (IRF) in which two spaceships are traveling in opposite directions at nearly the speed of light and one of them sends a light signal to the other one, the light signal travels at c relative to the IRF, not relative to either spaceship. That is why it has no trouble reaching the other spaceship but it will take a very long time according to the IRF. However, since the spaceships are traveling at a very high speed in the IRF, their clocks are running very slowly compared to the coordinate time of the IRF and so it will not take as long for them for the light to traverse the distance between them.

Think of speeds relative to an IRF, not relative to observers and you'll have no problem. After you set up a scenario in a given IRF, you can then transform the coordinates of the significant events defined according to that IRF into another IRF moving with respect to the first IRF and see how light also travels at c in the new IRF and produces all the same observables and measurements for each observer as the first IRF did.

This is the correct way to explain to newbies how Special Relativity describes what is happening.

 

[ghwellsjr]

I've drawn a diagram to show what happens in an IRF with two spaceships traveling in opposite directions at 0.8c. They both leave the blue spacestation at time 0 when they all set their clocks to 0. The dots mark off one-minute intervals for each spaceship and the spacestation. At 0.8c, gamma is 1.6667 so the dots for the spaceships are farther apart by that amount compared to the coordinate time. After one minute, the red spaceship sends a signal to the black spaceship at the speed of light according to the IRF. The black spaceship receives this signal when his clock reads 9 minutes even though it took over 13 minutes according to the IRF:

 

[K^2]

It might be useful to have the same diagram in IRF of each of the ships as well.

 

[ghwellsjr]

It might be useful to have the same diagram in IRF of each of the ships as well.

Ok, here's the one for the black spaceship's IRF:

Notice that the blue spacestation is now traveling at -0.8c and the red spaceship is traveling at -0.9756c. The time dilation for the spaceship is 1.6667 and for the red spaceship is 4.5556. The light signal, emitted by the red spaceship at its one-minute mark, travels in this IRF at c to the black spaceship at its nine-minute mark, just like in the original IRF.

And here's the one for the red spaceship's IRF:

The blue spacestation is traveling at 0.8c and the black spaceship is traveling at 0.9756c. The time dilation for the spaceship is 1.6667 and for the black spaceship is 4.5556. The light signal, emitted by the red spaceship at its one-minute mark, travels in this IRF at c to the black spaceship at its nine-minute mark, just like in the other two IRF's.

We could also pick any other IRF to show all the same events and they would all be in agreement. There is nothing special about any particular IRF, even one for which any observer is at rest--what we call its own IRF. That's the whole point of Special Relativity.

 

[K^2]

We could also pick any other IRF to show all the same events and they would all be in agreement. There is nothing special about any particular IRF, even one for which any observer is at rest--what we call its own IRF. That's the whole point of Special Relativity.

Of course. This just happens to be a very nice illustration to point to. It shows, visually, how the velocities add, how the times of events are different in each coordinate system, and how despite that, the proper time for each intercept remains the same. It'll be nice to point to if anybody else asks related questions. Thanks for taking the time.

 

[bobc2]

Ok, here's the one for the black spaceship's IRF:

Notice that the blue spacestation is now traveling at -0.8c and the red spaceship is traveling at -0.9756c. The time dilation for the spaceship is 1.6667 and for the red spaceship is 4.5556. The light signal, emitted by the red spaceship at its one-minute mark, travels in this IRF at c to the black spaceship at its nine-minute mark, just like in the original IRF.

And here's the one for the red spaceship's IRF:

The blue spacestation is traveling at 0.8c and the black spaceship is traveling at 0.9756c. The time dilation for the spaceship is 1.6667 and for the black spaceship is 4.5556. The light signal, emitted by the red spaceship at its one-minute mark, travels in this IRF at c to the black spaceship at its nine-minute mark, just like in the other two IRF's.

We could also pick any other IRF to show all the same events and they would all be in agreement. There is nothing special about any particular IRF, even one for which any observer is at rest--what we call its own IRF.

Great job witht the space-time diagrams, ghwellsjr!

That's the whole point of Special Relativity.

That may be a little bit of an overstatement, but the intended spirit of it works fine, and you did make very good points.

 

[Vandam]

Gwelsjr,
I really appreciate your effort drawing diagrams. Good job.
Unfortunately I am not too happy with your text.
The way you explain it is as if the time dilation occurs due to the further dots pacing of clockticks on the clock worldline.
There is also a danger of interpreting your diagrams as if timedilation happens because of the taking into account of the lightbeams.
Let me put it this way:
You have drawn three different diagrams, but to see how timedilation works it is better to look at one diagram only with more information on it. I did it for you:
(For simplicity I omited what red says):
Blue notices black (and red) time dilation because when his blue clock ticks 15 (not 13 as you wrote;)), in his BLUE frame (=3D spaceworld) the black and red clock show 9.
So far so good.
Now what black says:
Black notices time dilation of blue time because "when his black clock ticks 9 seconds, IN HIS BLACK FRAME (=3D spaceworld) the blue clock is only at 5,399 sec. (9 / 1,6667).
And still IN HIS BLACK 3D world the blue 'time dilation' means: red clock is only at 1,975 sec (9 / 4,5556).
Here you see clearly that the 'slowing down' of the blue and red clock have NOTHING to with the further spacing of blue or red dots relative to the black dots (red dots are equally spaced as black, blue dots are less spaced relative to the black ones!).

On a Loedel diagram (I'm now too lazy too make one) you would see that the time dilation has nothing to do with the further spacing of clock ticks on the worldline of a clock. Time dilation is about relativity of simultaneity. (And that's only possible if you consider all events out there as observer independent entities time indications included (block universe).

Just in case you wonder what the reciprocal time dilation is for blue at 9 (see sketch below):
in blue 3D space black clock is at 5,389.
In blue 3D space when blue clock is at 5,399 the black clock is at 3,238 (= 5,399 : 1,667 ).
3D cuts through 4D block spacetime!