Crossfire in an accelerating railway carriage.

Let's have a railway carriage.
The carriage is accelerating at a m/s/s
We fire a bullet across the carriage.
Einstein claimed acceleration and 'gravity' were the same thing so at first sight this suggests Newton and Einstein would expect the bullet to hit the same point on the other side of the carriage.

My first thought is that a clock on the other side of the carriage would be slightly behind 'our' clock. So the other side of the carriage won't have been accelerating for as long as 'our' side. Obviously if it has been accelerating for less time it won't be moving quite as fast as 'our' side. So using relativity to work out where the bullet lands isn't going to be quite as simple as using Newton's laws of physics.

 

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In the UK our carriages are longer than they are wide. The locomotive pulls from the long (front or back) and 'across' is the short distance from one side to the other.

-----Their Clock--------------------------------------------------------->>a
^
v
^
-----Our Clock----------------------------------------------------------->>a

What follows "I don't think so" doesn't (at first sight) seem to explain why Newton and relativity might give different answers. Or do you think they do give the same answer? Should we ignore what clocks tell us in favour of a Newtonian Universe?

 

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Someone with better understanding will correct me if needed, but it seems to me that the clocks on either side of the carriage are in synchrony, since they are in the same frame.

 

When you say the "same point" I'm assuming that mean the Newton and Einstein predict the same result, and not that the bullet will hit a point at the same distance from the front of the carriage as it was fired(which is what I believe that James R thought you meant by "same point".

Why? Nothing in Relativity predicts that clocks directly across from each other in the carriage would read different times. Clock's at different distances from the front of the train, yes, but not clocks equal distances from the front.

 

Let the width of the carriage be w and the speed of light be c. If a clock on 'our' side shows t then we'll see the other clock reading t-t_w where t_w=w/c . If 'our' side starts to accelerate at t0 does the other side start at t0-t_w or when their clock reads t0 ... at which time our clock reads t0+t_w. If the other side started later and accelerates at the same rate ... is it going at the same speed as our side?

 

You are in the same accelerating frame as both clocks, if they and you are perpendicular with respect to the acceleration. The bullet may have a longer flight time in the accelerated frame, but the clocks will be in agreement whether accelerated or stationary.

Oh, someone correct me, please. I know I got something wrong...

 

Just because you "see" the other clock reading an earlier time than your own doesn't mean that it actually reads a different time than yours or started accelerating after you did. This seems to be a common confusion among many when they start to learn about Relativity. If you want to know what time it is on the other clock "now", you would add t_w to the reading you see on it. Relativistic effects like time dilation and the Relativity of Simultaneity are what are left over after you have accounted for the propagation delay.

 

Hi. I'm yes and no about that. Imagine we sent a clock to (close to) the surface of the Sun. It shows Earth_now - 8minutes. Are we in orbit round the Sun as (and where) it is Earth_now -8 minutes or should we add 8 minutes and hope everything carries on as usual?

 

It's a different model, though. John, you have two clocks, one on either side of the carriage, and you fire a bullet from one side to the other. So what?

The clocks and the bullet are accelerated in two directions, but we can choose to ignore the earthward component for the purpose of this discussion.

Why do you see a problem or discrepancy again, please?

 

To be honest I don't. I do know that Einstein and Newton differed in their prediction of the deflection of starlight seen during an eclipse. I'm fishing for a qualitative explanation for the discrepancy. The current thread may not produce an answer or may be ill-conceived - in the event of either I'll have a think and come back later with (hopefully) a better stab at finding the reason.

 

With the starlight we are dealing with gravity, which involves curved space-time, which accounts for the difference between Newton and Einstein in this case. With the carriage, space-time is flat (assuming no gravity field) and we don't get that additional effect.

You add the eight min to the time you see on the Sun clock to get the time on it compared to your clock on Earth. However, that doesn't mean that your clock and the Sun clock will tick at the same rate. So for instance, if at some moment you see a reading on your clock of 12:00:00 and 11:52:00 on the Sun clock you know that at that moment, it s is 12:00:00 on the sun clock. However, 24 hrs later, when you again read 12:00:00 on your clock, you will read 11:51:59.82 on the Sun clock. Adding the same 8 mins again you get 11:59:59.82 for the actual time on the Sun clock. Every 24 hrs, the Sun clock losses another 0.18 sec compared to your own. We don't worry about the 8 min, as that remains a constant, we only consider the part that accumulates.

With the carriage, you see the clock directly across from you reading 12:00:00- t_w when you see your clock read 12:00:00. 24 hrs later you still read it as 12:00:00-t_w. The difference never increases or decreases. Thus by adding t_w to the reading you see you always get the same reading as your own clock. There is no accumulating difference and thus no time rate difference between the clocks.

 

What the hell?

 

With the accelerating carriage...you have to do work to get from one side to other - that is to say Clock A is at a lower potential than Clock B and Clock B is at a lower potential than Clock A. I have to admit I don't see the immediate consequences of this - but consequences there must be.

Edit... or maybe not. Maybe they are just peaks at the same potential.

 

No work is done crossing the carriage. Imagine you have a friction-less track crossing the carriage that you can roll a ball. If you start the ball rolling, it will roll across at a constant speed just like it would roll across a similar track on the surface of the Earth. If there were a difference in potential, the ball would lose speed as it rolled or you you have to keep applying a force to keep it moving at a constant speed. Saying that A is lower than B in potential while B is lower in potential than A is like saying you have a straight ramp that is "uphill" in both directions. ( And despite what your grandpa may have told you about hike to school, "uphill both ways" is not possible.)

 

I'd like to go a bit more extreme on the acceleration front. While the ball is rolling across the carriage continues to accelerate - the side the ball arrives at isn't going at the same speed as the side it was launched from. To hit the same* point on the other wall Newton suggests it will need to be going faster by (roughly) at_ball where t_ball is the time the ball takes to cross. So yes I think you do need to do work to get the ball to the same point on the other wall.

*same point being defined as points marked on both walls (with no acceleration) which are perpendicular to each other. The sides of the carriege being opposite side of a rectangle.

 

Your confusing yourself by jumping between frames. In the frame that the carriage is accelerating with respect to, yes, work is done on the ball, but that work is being done by whatever is accelerating the carriage. If the carriage is accelerating in the x direction, it is the work needed to change the x velocity component of the ball. The y component velocity you impart on the ball is totally separate from this. It remains constant, and once you get the ball rolling no additional work is needed to keep it rolling across the carriage. So while the whatever is accelerating the carriage doe do work on the ball, it is no more than the work it would done on the ball if it had not crossed the carriage. Thus if you have two balls and one stays put while the other crosses the width of the carriage, they both will have had an equal amount of the work done on them in the time it takes the second ball to cross the carriage.

In the frame of the accelerating carriage you just consider the apparent force towards the rear of the carriage acting on the ball. As long as the ball is moving perpendicular to this force, no work is needed to maintain the movement.

 

I'm certainly not suggesting this is all that should be considered. Different experiments will give different insights. As already mentioned, the insight I seek is why Newton and Einstein would disagree about the deflection of a beam of light in a gravitational field. Any port in a storm.

Your ball on a frictionless track looks (at first sight) to be usefully 'inertial' relative to the source and destination. However by constraining the path to start at A (one end of the track) and end at B (the other end of the track) I suspect Einstein and Newton would be forced into agreement about the result.

I may (often) be wrong but in my humble opinion the ingredients for a disagreement between Einstein and Newton are present in the OP (barring mistakes) it is 'just' a matter of identifying the assumption made by each that lead to different conclusions about (in particular) the point on the other wall where the bullet will land.

 

From https://en.wikipedia.org/wiki/Tests_of_general_relativity

So in 1915 Einstein noticed something he had missed in 1911.

So what did he miss?

A 'thought'...

In the accelerating carriage of the OP we have (or arguably don't have) a velocity difference between the two sides. IF there is a velocity difference there will be time dilation. If there is time dilation then it won't just be at a point in space - it will be in spacetime. (Compare Pound Rebka https://en.wikipedia.org/wiki/Pound–Rebka_experiment)