Crossfire in an accelerating railway carriage.

[Confused2]

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.

 

[James R]

Let's have a railway carriage.
The carriage is accelerating at a m/s/s
We fire a bullet across the carriage.

You mean parallel to the acceleration direction or perpendicular to it?

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.

I don't think so. Suppose we fire the bullet across from the right-hand side of the carriage to the left, at an initial angle of 90 degrees to the wall, as the train accelerates forwards. If you're inside the train, you will see the bullet strike the left-hand wall to the rear of the point on the right-hand wall where it was fired from. You could explain this as being due to the action of a mysterious, unexplained force pulling the bullet-in-flight towards the rear of the carriage (and you'd feel a similar force yourself). You might equally say that somebody somehow switched on a gravitational field that made objects 'fall' towards the rear.

 

[Confused2]

You mean parallel to the acceleration direction or perpendicular to it?

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

I don't think so. Suppose we fire the bullet across from the right-hand side of the carriage to the left, at an initial angle of 90 degrees to the wall, as the train accelerates forwards. If you're inside the train, you will see the bullet strike the left-hand wall to the rear of the point on the right-hand wall where it was fired from. You could explain this as being due to the action of a mysterious, unexplained force pulling the bullet-in-flight towards the rear of the carriage (and you'd feel a similar force yourself). You might equally say that somebody somehow switched on a gravitational field that made objects 'fall' towards the rear.

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?

 

[Dr_Toad]

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.

 

[Janus58]

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.

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".

My first thought is that a clock on the other side of the carriage would be slightly behind 'our' clock.

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.

 

[Confused2]

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?

 

[Dr_Toad]

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...

 

[Janus58]

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?

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.

 

[Confused2]

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.

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?

 

[Dr_Toad]

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?

 

[Confused2]

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.

 

[Janus58]

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.

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?

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.

 

[Dr_Toad]

What the hell?

 

[Confused2]

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.

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.

 

[Janus58]

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.)

 

[Confused2]

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

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.

 

[Janus58]

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.

 

[Confused2]

We fire a bullet across the carriage.

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.

 

[James R]

Confused2:

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.

In considering your original example of a bullet fired across the train, I don't see any reason why Einstein and Newton would disagree on the outcome - i.e. the position on the train carriage where the bullet will hit the wall. They would disagree on the interpretation of what happened, from inside the train carriage (though they would agree on the explanation from the inertial frame of a person standing next to the tracks watching the train accelerate). From the perspective inside the train, Newton would not be able to offer an explanation other than some kind of mysterious 'inertial' force acting on the bullet in flight, with no apparent cause. Einstein, on the other hand, could identify this as equivalent to a gravitational force, because for Einstein gravity is just an effect of working in a non-inertial frame.

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.

Yes, they would agree in this case, too. And, by the way, to go 'straight across' the carriage on its track, the ball would have to be subject to a force directed in the direction of acceleration of the train. This is for the reason you gave: that by the time it reaches the 'far' wall, it must be going faster than it was when it left the 'near' wall. That is, it must accelerate, and that requires a force. Again, though, in the non-inertial frame of the carriage as its accelerates, Newton would describe this force as a pseudoforce, whereas Einstein would (again) describe it as a gravitational 'force'.

When it comes to an experiment with light rather than bullets, we discover something interesting. In the accelerating frame of the carriage, we need to account for the fact that light fired across the carriage hits the wall 'behind' where we would expect if the carriage was stationary. Newton really can't explain this, as he doesn't address the question of (pseudo-)forces on light. Einstein would like to explain the result as a gravitational one, as in the case of the bullet, but recall that light has no mass. To be consistent, Einstein must conclude that light, despite having no mass, is nevertheless affected by gravity. And it is this insight, in part, that drives us towards the curved-spacetime picture of general relativity.

 

[Confused2]

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

Henry Cavendish in 1784 (in an unpublished manuscript) and Johann Georg von Soldner in 1801 (published in 1804) had pointed out that Newtonian gravity predicts that starlight will bend around a massive object.[13][14] The same value as Soldner's was calculated by Einstein in 1911 based on the equivalence principle alone. However, Einstein noted in 1915 in the process of completing general relativity, that his (and thus Soldner's) 1911 result is only half of the correct value. Einstein became the first to calculate the correct value for light bending.[15]

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)