Force due to acceleration and time flowing differently

[PeroK]

But I now understand that that is not possible or the answer is already given probably but too detailed for me?

It's the opposite. It's simpler. Proper time is the length of the path through spacetime. You embellish this with concepts such as "time slows down because of velocity/acceleration/force", which just confuses the issue.

 

[HansH]

I can produce scenarios where this is not true. You can have double the acceleration with the same difference between clocks.

I talked about acceleration during a certain time interval. so not ony the acceleration but also the duration of the accelleration. so it is about the combination of acceleration during a certain time interval (causing a speed difference between the 2 rockets) and the building up rate of the difference between the clocks. of course the difference in clocks builds up over the time that the speed difference remains, so the interval of keeping th speed difference before going back to same speed for both rockets also is in the relation. So I would expect you could write this down in a total equation something like :

difference in clocks= k1 x F1(build up speed difference) x F2(duration that the speed difference remans)

difference in clocks= k1 x F3(accelaration) x F4(duration the acceleration) x F2(duration that the speed difference remans)
so the question is: can we define such a relation where k1 is a constant and if so why does k1 then have the value as is has?

 

[Nugatory]

I cannot prevent that the question pops up in my head at the wrong moment. So I hoped for a clear high level answer already from people who are already comfortable with the matter.

However there is no answer that you will recognize as clear because, as this thread demonstrates, you lack the necessary background to follow a high-level answer.

There is also a problem with jumping into any question that just “pops up in [your] head”: as I said in your other thread

That’s a haphazard and disjointed activity that is unlikely to ever lead to a coherent understanding.
Taylor and Wheeler’s “Spacetime Physics” is available free online. Start at the first page. We can help you over the hard spots.

….and well before you get to the end, you’ll be able to handle this sort of question.

 

[HansH]

My preferred non-inertial reference frame is the radar coordinates which were popularized by Dolby and Gull. One thing that is nice about radar coordinates is that there is a known procedure for determining the metric given only the proper acceleration of the non-inertial observer. This procedure is a concrete answer to your question about the link between acceleration and the metric (which includes the "flow of time")

I have linked to two simple papers on this topic. I would recommend that you read both of them. If you cannot understand these papers then you are probably not yet ready for this question and should focus on the basics first.

This is indeed a no go for me. it already starts with fig 2 where it is not indicated what is on the axis and a lot of terms I am not familiar with : hypersurfaces of simultaneity, x µ = x µ (γ) (τ) etc.

 

[jbriggs444]

difference in clocks= k1 x F1(build up speed difference) x F2(duration that the speed difference remans)

Yes, the Lorentz transform contains a formula somewhat like that. $$t'=\gamma(t-\frac{vx}{c^2})$$
Your "duration the speed difference remains" is encoded in $x$"
Your "build up speed difference" is encoded in $v$ and $\gamma$.

The take away I get from this formula is that "There is a systematic synchronization skew based on displacement in the direction of relative travel. The higher the relative speed, the greater the skew".

Others take away a pithier form: "leading clocks lag".

 

[HansH]

Yes, the Lorentz transform contains a formula somewhat like that.

yes thanks, that helps. it is half of the discussion about the part including speed difference during a certain time. The other part should then be connected to the effects of building up this speed difference by an acceleration during a certain interval to build up the speed difference. and that results in the force in the other part of the equation I showed. So based on that both parts could be calculated so then it should be possible (at least for a physicist) to write down the complete equation. not sure if the part about building up the gamma factor during the acceleration phase. and then the k1 factor should follow.

 

[PeterDonis]

can we define such a relation

Only for the particular kind of scenario you are describing. It won't generalize to other scenarios. So, as I've already pointed out in post #66, the question is whether you want to learn a particular set of rules that will work in this particular scenario, but won't generalize, or whether you want to learn the general rules that will work for all scenarios. Which is it?

 

[PeroK]

yes thanks, that helps. it is half of the discussion about the part including speed difference during a certain time. The other part should then be connected to the effects of building up this speed difference by an acceleration during a certain interval to build up the speed difference. and that results in the force in the other part of the equation I showed. So based on that both parts could be calculated so then it should be possible (at least for a physicist) to write down the complete equation. not sure if the part about building up the gamma factor during the acceleration phase. and then the k1 factor should follow.

This is all wrong thinking - as has already been pointed out. Ultimately, differential ageing depends only on the velocity profiles in any IRF. Yes, velocity profiles can be deduced from proper acceleration profiles and hence from force profiles, but that's hardly the point. It's like trying to work out whether someone is speeding by looking at fuel consumption rather than the speedometer.

You are, of course, free to plough your own furrow. You are not the first and no doubt you won't be the last; but, it won't lead to an understanding of SR.

 

[PeterDonis]

Ultimately, differential ageing depends only on the velocity profiles in any IRF.

I would put this a bit differently: ultimately, differential aging depends only on the arc length along worldlines. In a situation where an inertial frame can be defined that covers the entire scenario, the arc length along worldlines can be computed as a function of the velocity profiles of the worldlines in that frame. The acceleration does not appear anywhere in this computation.

If one only knows the acceleration, one can of course compute the velocity profiles from the acceleration profiles, but that requirement would be specific to that particular scenario.

Also, not all scenarios in relativity have the property that an inertial frame can be defined that covers the entire scenario; all scenarios in special relativity do (since in SR there are always global inertial frames), but many scenarios in general relativity do not.

 

[HansH]

I can produce scenarios where this is not true. You can have double the acceleration with the same difference between clocks.

I talked about acceleration during a certain time interval. so not ony the acceleration but also the duration of the accelleration. so it is about the combination of acceleration during a certain time interval (causing a speed difference between the 2 rockets) and the building up rate of the difference between the clocks. of course the difference in clocks builds up over the time that the speed difference remains, so the interval of keeping th speed difference before going back to same speed for both rockets also is in the relation. So I would expect you could write this down in a total equation something like :

difference in clocks= k1 x F1(build up speed difference) x F2(duration that the speed difference remans)

difference in clocks= k1 x F3(accelaration) x F4(duration the acceleration) x F2(duration that the speed difference remans)
so the question is: can we

Only for the particular kind of scenario you are describing. It won't generalize to other scenarios. So, as I've already pointed out in post #66, the question is whether you want to learn a particular set of rules that will work in this particular scenario, but won't generalize, or whether you want to learn the general rules that will work for all scenarios. Which is it?

For understanding the general rules you need a 5 years physics study I suppose? That is at least not realistic in combination with my current situation in life. so yes good to learn the general rules, but probably not reachable. That I should have done then 30 years ago but would have given other compromises then also. But that is at least one thing I know from relativity that you cannot go back in time to make other choices. I was simply interested in the answer if there could be such a relation as I suspected, but that that turned out to be that difficult and probably not exsisting was not what I suspected.

 

[PeterDonis]

For understanding the general rules you need a 5 years physics study I suppose?

Go read post #70, where I stated a simple general rule that should not require 5 years of physics study to grasp, just a basic grasp of geometry.

 

[HansH]

post 70 refers back to post where you conclude: '
'the difference between the watches is due to the difference in arc length along the two worldlines between the two meeting events"
when I look what an arc lengt is that is the length of a trajectory in 4D spacetime as I understand but what do you mean by 2 meeting events? Isuppose you mean the trajectory of 2 observers that had different accelerations. but what is then a meeting event? and then the message you want to bring over from that senstence is still not clear to me in relation to different times on watches and force due to acceleration.
so what I now know is not sufficient to get the message from your general rule I suppose.

 

[anuttarasammyak]

ok but you can calculate the potential energy from the force working over a distance, so that makes the question not basically different I suppose.

Let us see the simple case of rotation. In the IFR where the axis of rotation is at rest, as SR is applicable, a rotating body undertakes time dilation of ratio
$$\gamma^{-1}=\sqrt{1-\frac{v^2}{c^2}}=\sqrt{1-\frac{r^2\omega^2}{c^2}}$$
where r is radius or rotation and $\omega$ is angular velocity.
In the non IFR system of the rotating body this ratio is interpreted as
$$\sqrt{g_{00}}=\sqrt{1+\frac{2\phi}{c^2}}$$
where
$$\phi=-\frac{1}{2}r^2\omega^2$$
, potential energy of centrifugal force per unit mass. A crew of the rotating whose system is non IFR observes that his lower potential energy than the center of rotation makes his time dilation.

 

[PeterDonis]

when I look what an arc lengt is that is the length of a trajectory in 4D spacetime as I understand

Yes.

but what do you mean by 2 meeting events?

The two events (points in spacetime) where the two observers meet and compare their watches. At the first event they ensure that their watches are synchronized; at the second event they compare the elapsed times on their watches and (in your scenario) find that they are different. The worldlines of the two observers must intersect that both of these points, so the difference in their elapsed times is just the difference in the arc lengths of their worldlines between the two points.

 

[HansH]

Yes.The two events (points in spacetime) where the two observers meet and compare their watches. At the first event they ensure that their watches are synchronized; at the second event they compare the elapsed times on their watches and (in your scenario) find that they are different. The worldlines of the two observers must intersect that both of these points, so the difference in their elapsed times is just the difference in the arc lengths of their worldlines between the two points.

ok clear. that is an easy to understand conclusion. (although I don't understand why but that is probably where the 5 years is for) but then given that difference in arc length, the next part of the story is about the question what happens during the sequence where both observers follow their trajectory while a certain acelleration as function of the position on each trajectory holds in order to make it possible to follow that trajectory and then the question if we can say something about that in order to determine a relation between that acelleration and the build up of the difference in stopwatches time. That is the most general way of looking to that problem I suppose?

 

[PeterDonis]

I don't understand why

Meaning you don't understand why the arc lengths are different? That's easy: because you are taking two different paths through spacetime between the same points. It's just geometry: it's the same as if you had two cars that took two different routes between, say, New York and Los Angeles. You would expect the elapsed odometer readings on the cars to be different because the routes were different. This is just the same thing in spacetime.

the next part of the story is about the question what happens during the sequence where both observers follow their trajectory while a certain acelleration as function of the position on each trajectory holds in order to make it possible to follow that trajectory

An object's trajectory through spacetime is determined by the geometry of spacetime and the object's 4-velocity. Proper acceleration is only relevant because it changes the object's 4-velocity.

in order to determine a relation between that acelleration and the build up of the difference in stopwatches time.

That relation will be different for different scenarios. In fact there are even scenarios in GR (in curved spacetime) where two objects that both have zero proper acceleration for all time have different elapsed times between two meetings.

That is the most general way of looking to that problem I suppose?

No. The most general way of looking at things is to look at spacetime geometry and object trajectories, and not to even bother about trying to find scenario-specific relationships between acceleration and other things, or even between velocity and other things.

Again, consider the analogy to two cars that take two different routes between New York and Los Angeles. The obvious general way of looking at such scenarios is to look at the geometry: the distance along each route. You could try to concoct relationships between that and, say, the compass direction in which each car points as a function of time along its route, or the rate of change of that compass direction, but any such relationship would be specific to the scenario and wouldn't really tell you anything useful.

 

[HansH]

Meaning you don't understand why the arc lengths are different?

no that is clear. what is unclear is how I can understand how that relates to difference between the watches at the meeting point and specifically how much time difference compared to arc length difference..

 

[PeterDonis]

is unclear is how I can understand how that relates to difference between the watches

The arc length difference is the difference between the watches. They're the same thing. Clocks record arc length along their worldlines in spacetime, just as odometers record arc length along their paths in ordinary space.

 

[HansH]

In fact there are even scenarios in GR (in curved spacetime) where two objects that both have zero proper acceleration for all time have different elapsed times between two meetings.

What do you mean by 'proper accelleration' When i am in freefall does that mean my proper accelleration is zero? so what for example about falling in freefall from the top of a building to earth. what is then my proper acceleration? is it then 0 or 1g?

 

[Dale]

This is indeed a no go for me. it already starts with fig 2 where it is not indicated what is on the axis and a lot of terms I am not familiar with : hypersurfaces of simultaneity, x µ = x µ (γ) (τ) etc.

Then I would recommend tabling this question for now and focusing on the basics. Do you understand Lagrangian mechanics? That will be generally helpful in learning physics.

Then the next thing to focus on is basic special relativity, especially four-vectors and the geometric interpretation of special relativity.

 

[PeroK]

Then the next thing to focus on is basic special relativity ...

... like the first postulate?

 

[HansH]

Do you understand Lagrangian mechanics?

I don't think that was part of my study somewhere.

 

[PeterDonis]

What do you mean by 'proper accelleration'

The acceleration measured by an accelerometer attached to the object.

When i am in freefall does that mean my proper accelleration is zero?

Yes.

 

[Dale]

I don't think that was part of my study somewhere.

If you want something useful, nothing you can study in relativity will be more useful than learning Lagrangian mechanics. And when you do learn the Lagrangian approach it will eventually make studying non-inertial reference frames easier. However, it isn't as "exciting" a topic as relativity.

 

[HansH]

Yes.

ok that helps. Then it means that I am not referring to the tem ''proper acceleration' but to the accelleration as result of an applied force such as a rocket engine. But then I think you are going to say that an observer cannot discriminate between gravity in curved spacetime and a rochet engine in flat spacetime due to the principle of equivalence. But at least for the acceleration he could isolate the effect of a rocket when he knows he has his rocket on or off as this gives the same relation between acceleration mass and force. so you could probably calculate the worldline with and without external forces and calculate only the contribution due to the external force to the difference in time between the watches. at least that should be easy for situations with flat spacetime as there ony the rocket could cause a change in speed.

 

[HansH]

If you want something useful, nothing you can study in relativity will be more useful than learning Lagrangian mechanics. And when you do learn the Lagrangian approach it will eventually make studying non-inertial reference frames easier. However, it isn't as "exciting" a topic as relativity.

thanks I will keep in mind.

 

[Dale]

Then it means that I am not referring to the tem ''proper acceleration' but to the accelleration as result of an applied force such as a rocket engine.

That is proper acceleration. An accelerometer does detect the acceleration as a result of a rocket engine. It seems like you may have a Newtonian physics misunderstanding here.

 

[HansH]

Yes. that would probably mean that the main idea of the topic cannot be united with general relativity due to the equivalence principle that cannot isolate an external acceleration from the effect of curvature on the path of a worldline? also based on the earlier statement that 2 different worldlines can exist with proper acceleration=0 over the whole worldline.

 

[Dale]

Yes. that would probably mean that the main idea of the topic cannot be united with general relativity due to the equivalence principle that cannot isolate an external acceleration from the effect of curvature on the path of a worldline?

I understand all of the words you are using but not the way you are using them.

Proper acceleration is the acceleration measured by an accelerometer. Both GR and Newtonian physics can describe the acceleration measured by an accelerometer. The difference between GR and Newtonian physics is not regarding proper acceleration. It is that Newtonian physics considers gravity to be a real force that is just coincidentally undetectable by accelerometers while GR considers gravity to be locally a fictitious force that is undetectable just like all other fictitious forces.

 

[HansH]

I understand all of the words you are using but not the way you are using them.

Proper acceleration is the acceleration measured by an accelerometer. Both GR and Newtonian physics can describe the acceleration measured by an accelerometer. The difference between GR and Newtonian physics is not regarding proper acceleration. It is that Newtonian physics considers gravity to be a real force that is just coincidentally undetectable by accelerometers while GR considers gravity to be locally a fictitious force that is undetectable just like all other fictitious forces.

I assume I am not talking about the difference between GR and Newtonian physics but the difference between GR with curved spacetime and special relativity with only flat spacetime. probably it helps to draw the worldlines in special relativity with the 2 rockets I used earlier and then calculate the difference between the stopwatches and compare that with the summed up product (integral) of acceleration times the time that the accelaration took at a certain value so integral(a(t)dt) (or probably a more relativistic equivalent that I cannot produce now) and divide that integral by the calculated timedifference between the stopwatches and see if this is a constant or not. But I agree that this should also work in general and for GR because otherwise it is not a valid theory. But proving that a theory is wong only needs one example.

 

[PeterDonis]

Then it means that I am not referring to the tem ''proper acceleration'

Yes, you are. See below.

but to the accelleration as result of an applied force such as a rocket engine.

Such an "applied force" causes proper acceleration.

I think you are going to say that an observer cannot discriminate between gravity in curved spacetime and a rochet engine in flat spacetime due to the principle of equivalence.

Not at all. Remember, in relativity "gravity" is not a force. The force you feel when you stand at rest on the surface of the Earth is not "gravity". It's the force of the Earth's surface pushing up on you. Just as, if you stand at rest on the floor of a rocket accelerating at 1 g, the force you feel is the force of the rocket's floor pushing up on you. The principle of equivalence says that you cannot distinguish these two cases by local observations, but that is not the same as saying that you can't distinguish "gravity" from the effects of a rocket engine in flat spacetime.

Basically what the principle of equivalence is saying is that proper acceleration is proper acceleration, and by itself it doesn't tell you want kind of spacetime geometry the proper acceleration is occurring in.

the equivalence principle that cannot isolate an external acceleration from the effect of curvature on the path of a worldline?

There is no such thing as "the effect of curvature on the path of a worldline" if by "curvature" you mean "spacetime curvature". That is getting things backwards. It's not that you start out with a worldline, and then you put it in one spacetime geometry or another and see what happens to it. You have the spacetime geometry first, and then you look at the behavior of worldlines in it. There is no way to pick out "the same worldline" in two different spacetime geometries and compare the effects of one spacetime geometry vs. another on "the worldline". There is no way to even define a "worldline" at all independently of a spacetime geometry.

 

[PeterDonis]

I assume I am not talking about the difference between GR and Newtonian physics but the difference between GR with curved spacetime and special relativity with only flat spacetime.

If you want to put gravity in your scenario, yes, that would be the usual assumption in this forum since it is the relativity forum.

probably it helps to draw the worldlines in special relativity with the 2 rockets I used earlier and then calculate the difference between the stopwatches

If this means "calculate the difference in arc lengths of the worldlines", then yes, this is the general method that always works, whether spacetime is flat or curved.

and compare that with the summed up product (integral) of acceleration times the time that the accelaration took at a certain value so integral(a(t)dt) (or probably a more relativistic equivalent that I cannot produce now) and divide that integral by the calculated timedifference between the stopwatches and see if this is a constant or not.

I'm not sure what you are trying to accomplish with all of this. But whatever it is, as I have already pointed out several times now, it would only be applicable to this scenario and nothing it might tell you would generalize usefully to other scenarios.

 

[Dale]

the difference between GR with curved spacetime and special relativity with only flat spacetime

I am not sure that there is any reasonable way to compare the lengths of two worldlines in different spacetimes. I mean, you can certainly compare their lengths, but trying to say the difference in length was caused by something seems impossible. Two different spacetimes are causally disconnected by definition.

 

[HansH]

I can produce scenarios where this is not true. You can have double the acceleration with the same difference between clocks.

I talked about acceleration during a certain time interval. so not ony the acceleration but also the duration of the accelleration. so it is about the combination of acceleration during a certain time interval (causing a speed difference between the 2 rockets) and the building up rate of the difference between the clocks. of course the difference in clocks builds up over the time that the speed difference remains, so the interval of keeping th speed difference before going back to same speed for both rockets also is in the relation. So I would expect you could write this down in a total equation something like :

difference in clocks= k1 x F1(build up speed difference) x F2(duration that the speed difference remans)

difference in clocks= k1 x F3(accelaration) x F4(duration the acceleration) x F2(duration that the speed difference remans)
so the question is: can we

 

[HansH]

I'm not sure what you are trying to accomplish with all of this. But whatever it is, as I have already pointed out several times now, it would only be applicable to this scenario and nothing it might tell you would generalize usefully to other scenarios.

The point is that mass curves spacetime and mass also resists against a change in speed. and a change in speed is acceleration and aceleration also plays a role in curvature of spacetime. So as I assume these are both properties of mass, it could well be that there is an underlying common reason for both properties to behave like this that we possibly cannot see because it could be in the machinery behind nature that we cannot see. so if this is the case there should be some relation so that was where I was looking for and if this was probably already known in the physics community.