Combined speeds at relativistic velocities

[Moontanman]

If two space craft were approaching each other at 99.999% of the speed of light, what would the speed of each appear to be to the other since the speeds would not add up like two trains traveling at 100 mph toward each other. 

Edited May 1 by Moontanman

[exchemist]

3 minutes ago, Moontanman said:

If two space craft were approaching each other at 99.999% of the speed of light, what would the speed of each appear to be to the other since the speeds would not add up like two trains traveling at 100 mph toward each other. 

99.999% of c relative to what, though? 

[pzkpfw]

Edit 2: darn, misread. I'll leave this here but as swansont points out I've misread your question.

[

The formula is here:

https://en.wikipedia.org/wiki/Velocity-addition_formula

The "interesting" bit (to me) is the division by  1 + ( ( v x u' ) / ( c x c ) )

Where v and u' are small fractions of c, this ends up being a division by almost 1, i.e. at regular day to day speeds, it hardly makes a difference.

But where v and u' are large fractions of c, that becomes close to a division by two, i.e. you can add to speeds close to c, and still get a speed close to c.

As exchemist points out, you need to be careful who thinks what speed is what.

edit: but don't mix this up with closing speed, that can be as much as 2c.

e.g. someone considering themselves as still, who sees two rockets coming towards them from opposite directions, each at 99% of c, does see the gap between the rockets decreasing at close to 2c. But that's not the same as seeing a thing moving faster than c; neither that middle observer nor either rocket sees any rocket going over c here.

]

Edited May 1 by pzkpfw

[swansont]

19 minutes ago, Moontanman said:

If two space craft were approaching each other at 99.999% of the speed of light, what would the speed of each appear to be to the other since the speeds would not add up like two trains traveling at 100 mph toward each other. 

.99999 c

You’ve stated the answer already. Each would see this.

The velocity addition formula is for two speed relative to something else. e.g. two trains traveling at 50 mph relative to the ground (or e.g. 70 mph and 30 mph) would approach each other at 100 mph (the relativistic correction would be negligible)

 

[Moontanman]

So there would be no combined speed effect at all? 

[pzkpfw]

Just now, Moontanman said:

So there would be no combined speed effect at all? 

Try rephrasing the question with a little more precision on who is measuring what.

[Moontanman]

5 minutes ago, pzkpfw said:

Try rephrasing the question with a little more precision on who is measuring what.

They are measuring each others speed.

 

[pzkpfw]

Then swansont gave you the answer.

[exchemist]

5 minutes ago, Moontanman said:

So there would be no combined speed effect at all? 

You can only define a speed relative to some other object. So far the only objects you have mentioned are the two spacecraft. Is 99.999% of c measured relative to the other spacecraft or to something else? If the former, you have already stated the answer. If the latter you need to say what that something is.

You could for example say each space craft approaches the same space station, from diametrically opposite directions, at 99.999% of c relative to the space station.   

[Moontanman]

1 minute ago, exchemist said:

You can only define a speed relative to some other object. So far the only objects you have mentioned are the two spacecraft. Is 99.999% of c measured relative to the other spacecraft or to something else? If the former, you have already stated the answer. If the latter you need to say what that something is.

You could for example say each space craft approaches the same space station, from diametrically opposite directions, at 99.999% of c relative to the space station.   

Would the space station influence the speed the two space craft measure on each other? 

[exchemist]

Just now, Moontanman said:

Would the space station influence the speed the two space craft measure on each other? 

OK I give up. 

[Moontanman]

5 minutes ago, exchemist said:

OK I give up. 

Evidently I am too slow to get this one. I understand the two space craft would be approaching the space station at .9999 c but what would they measure each others speed at?

 

Edited May 1 by Moontanman

[pzkpfw]

4 minutes ago, Moontanman said:

Would the space station influence the speed the two space craft measure on each other? 

The existence of the space station doesn't affect anything. But if that's what speeds are being measured against, it changes your scenario.

You started off saying each rocket measures the other as going some speed. Well then that's just what they measure.

But if you change the scenario so that each rocket measures some speed from themselves to the station, and the other rocket as at some speed relative to the station, then yes, each rocket needs to use the proper formula to determine the speed of the other rocket relative to themselves.

[Moontanman]

2 minutes ago, pzkpfw said:

The existence of the space station doesn't affect anything. But if that's what speeds are being measured against, it changes your scenario.

You started off saying each rocket measures the other as going some speed. Well then that's just what they measure.

But if you change the scenario so that each rocket measures some speed from themselves to the station, and the other rocket as at some speed relative to the station, then yes, each rocket needs to use the proper formula to determine the speed of the other rocket relative to themselves.

Ok, so they couldn't measure their relative speeds without the station to compare it to?

I am trying guys

[pzkpfw]

Just now, Moontanman said:

Ok, so they couldn't measure their relative speeds without the station to compare it to?

No they can measure speed relative to themselves, as in your first post.

In your first post, rocket A (to give it a name) measures rocket B to be coming at 0.99999c.

And vice versa.

 

[Moontanman]

2 minutes ago, pzkpfw said:

No they can measure speed relative to themselves, as in your first post.

In your first post, rocket A (to give it a name) measures rocket B to be coming at 0.99999c.

And vice versa.

 

My thoughts on this is that the rockets would measure a combined speed but still less than c . Out side or before relativity the two space craft would have been thought to approach each other as faster than c but relativity precludes that so I thought they would see each other approaching at a speed faster than .9999 c... something like .9999999 c or something closer to c but not faster than c.

[pzkpfw]

You are confusing yourself by being imprecise. You cannot make your mind up on whose point of view is involved. Who is the observer, one of the rockets? Or someone else?

 

(And: As noted before, someone else (not in the rockets) can consider the rockets as approaching each other at more than c, closing speed can be as much as 2c.)

 

[Moontanman]

4 minutes ago, pzkpfw said:

You are confusing yourself by being imprecise. You cannot make your mind up on whose point of view is involved. Who is the observer, one of the rockets? Or someone else?

 

(And: As noted before, someone else (not in the rockets) can consider the rockets as approaching each other at more than c, closing speed can be as much as 2c.)

 

The observer is on each rocket, the rocket judges its own speed by its departure point, from my point of view it looks like the combined speeds would be greater than c, I understand this is not possible except in my frame of reference but what would the two space craft measure each others speed as? 

[joigus]

Moon, I think you're trying to think of an "impartial" observer who's sitting on some dock of the aeronautical bay, so to speak, and watches both approaching each other at 0.99999c. Then they would see each other approaching at higher than that, but never c or higher. You must run the Einstein velocity transformation formulas that  to see how much.

The relative speed from their POV would indeed be closer to c than that 0.9999c (or however many pieces of c she/he sees them from the dock.

If you actually run the calculations, you'd find, I don't know, something like 0.999999999999999999999999999999c relative to each other (you actually must run the calculation if you want to know how many 9 digits closer to c).

I think you're implicitly thinking of this "impartial observer" but failing to say so, and causing some amount of understandable confusion.

Is that so?

Does that help?

Edited May 1 by joigus
minor correction

[pzkpfw]

  

15 minutes ago, Moontanman said:

The observer is on each rocket, the rocket judges its own speed by its departure point, from my point of view it looks like the combined speeds would be greater than c, I understand this is not possible except in my frame of reference but what would the two space craft measure each others speed as? 

 

I think you really need to get clear on the difference between measuring closing speed (which is the speed of a gap, a nothing), and the speed of a thing.

And it depends a lot on who is observing.

Say the fastest car in the World can do 500 km/h. Stick two of them on a track facing each other and run them, at top speed.

The gap between (from the point of view of the track) them is decreasing at 1000 km/h. Hang on! That's faster than the car can go! But that 1000 km/h isn't the speed of either car according to the track. Sure, if the track considers it from the point of view of one of the cars, then the other is getting closer at 1000 km/h, but that reference point is moving according to the track. It's an illusion, if you like.

Relativistic addition comes in (in this scenario) when you consider the point of view from one of the cars. Each car can consider itself as still, and the other car moving towards it. But note that the track is also moving towards it! The track is moving towards each car at 500 k/h, and the other car is moving towards it at 500 km/h relative to the track.

And that's where you cannot (at relativistic speeds where it starts to matter) just add the 500 and 500. Each car will consider the other car approaching at 999.99999 km/h.

The track considers the closing speed as 1000 km/h, and the cars consider the other is approaching at 999.99999 km/h. These are different numbers.

Edited May 1 by pzkpfw

[exchemist]

37 minutes ago, Moontanman said:

Evidently I am too slow to get this one. I understand the two space craft would be approaching the space station at .9999 c but what would they measure each others speed at?

 

No, you've got it, that's the point. You obviously don't add the velocities, as you would in classical dynamics. 

[Moontanman]

8 minutes ago, joigus said:

Moon, I think you're trying to think of an "impartial" observer who's sitting on some dock of the aeronautical bay, so to speak, and watches both approaching each other at 0.99999c. Then they would see each other approaching at higher than that, but never c or higher. You must run the Einstein velocity transformation formulas that  to see how much.

The relative speed from their POV would indeed be closer to c than that 0.9999c (or however many pieces of c she/he sees them from the dock.

If you actually run the calculations, you'd find, I don't know, something like 0.999999999999999999999999999999c relative to each other (you actually must run the calculation if you want to know how many 9 digits closer to c).

I think you're implicitly thinking of this "impartial observer" but failing to say so, and causing some amount of understandable confusion.

Is that so?

Does that help?

Impartial observer... not what I had in mind, I was just thinking of the two rockets approaching each other and how they would see each other. 

8 minutes ago, pzkpfw said:

  

 

I think you really need to get clear on the difference between measuring closing speed (which is the speed of a gap, a nothing), and the speed of a thing.

And it depends a lot on who is observing.

Say the fastest car in the World can do 500 km/h. Stick two of them on a track facing each other and run them, at top speed.

The gap between (from the point of view of the track) them is decreasing at 1000 km/h. Hang on! That's faster than the car can go! But that 1000 km/h isn't the speed of either car according to the track. Sure, if the track considers it from the point of view of one of the cars, then the other is getting closer at 1000 km/h, but that reference point is moving according to the track. It's an illusion, if you like.

Relativistic addition comes in (in this scenario) when you consider the point of view from one of the cars. Each car can consider itself as still, and the other car moving towards it. But note that the track is also moving towards it! The track is moving towards each car at 500 k/h, and the other car is moving towards it at 500 km/h relative to the track.

And that's where you cannot (at relativistic speeds where it starts to matter) just add the 500 and 500. Each car will consider the other car approaching at 999.99999 km/h.

The track considers the closing speed as 1000 km/h, and the cars consider the other is approaching at 999.99999 km/h. These are different numbers.

The relative speeds of the cars was what I was getting at, I really didn't realize I wasn't being clear on that. I understand there is no track to measure anything by in relativity. 

[joigus]

1 minute ago, Moontanman said:

Impartial observer... not what I had in mind, I was just thinking of the two rockets approaching each other and how they would see each other. 

If you don't summon any other observer, then it's 0.99999c relative to each other, as Swansont and others said or implied, and/or/thus I'm missing the point.  /  / 🤷‍♂️

[swansont]

If two rockets each approached the space station at 0.5c, from opposite directions, they would be approaching each other at 0.8c

(.5 + .5)/(1 + 0.5*0.5)

(Galilean addition would give you c)

[pzkpfw]

For Moontanman: in the above, that 0.5c is as measured by the space station, and each rocket to the station; and the 0.8c is as measured by each rocket to the other rocket.

For the space station, the closing speed of the rockets (which is not the speed of either rocket) is c.

Edited May 1 by pzkpfw