Is there a maximum relativistic acceleration?

[Arend]

TL;DR

The relativistic length contraction expression should result in acceleration limitattion.

Hello, I am new to the forum. First, since now, a deep thank you for your patience with my amateur views. This question just popped up after doing the time derivative of the relativistic length contraction, and assuming that the maximum length change in time would be the light velocity. After considering that the object velocity is << light velocity, the result is that there is a maximum acceleration that is inversely proportional to the object size and velocity. Doing something wrong?

 

[Ibix]

There is no maximum acceleration in relativity.

Your attachments haven't worked, but you mention object size. Relativity does make it impossible to consider rigid objects under a limited range of circumstances (inertial motion, or constant and eternal rotation or Rindler acceleration, basically). It's possible you've tripped over that, and you may want to look up Bell's spaceships for a textbook demonstration of this.

 

[PeroK]

TL;DR Summary: The relativistic length contraction expression should result in acceleration limitattion.

Hello, I am new to the forum. First, since now, a deep thank you for your patience with my amateur views. This question just popped up after doing the time derivative of the relativistic length contraction, and assuming that the maximum length change in time would be the light velocity. After considering that the object velocity is << light velocity, the result is that there is a maximum acceleration that is inversely proportional to the object size and velocity. Doing something wrong?

The problem is that you cannot have the acceleration of two ends of an object starting simultaneously in two different frames of reference. SR is not just length contraction and time dilation - there is also the relativity of simultaneity.

Let's take an object of length $L$ at rest in some reference frame and consider two cases.

1) The object accelerates such that, in the original rest frame, both ends of the rod always have the same velocity. In this frame there is no length contraction, as the two ends remain a distance $L$ apart at all times.

Note that in this case, the object must be internally stretching. And, if the acceleration stops at some time, then in the new rest frame of the moving object it will have been stretched. In the rest frame of the rod, therefore, the front of the object must have been accelerating slightly more than the rear. It might be an good exercise to work this out fully.

2) If the object accelerates such that it retains it original length, then the acceleration of the two ends is not equivalent in the original rest frame. The rear of the object must have slightly more acceleration than the front in the original rest frame. So, it is not a rigid object moving at constant velocity. So, we do not have simple length contraction. Again, it would be a good exercise to work out the details fully.

 

[Arend]

There is no maximum acceleration in relativity.

Your attachments haven't worked, but you mention object size. Relativity does make it impossible to consider rigid objects under a limited range of circumstances (inertial motion, or constant and eternal rotation or Rindler acceleration, basically). It's possible you've tripped over that, and you may want to look up Bell's spaceships for a textbook demonstration of this.

Hi Ibix, fixed the attachments. Thank you for your comments. The point is that applying the length contraction expression for low velocity objects, would result in an acceleration limit.

 

[Orodruin]

Hi Ibix, fixed the attachments. Thank you for your comments. The point is that applying the length contraction expression for low velocity objects, would result in an acceleration limit.

You cannot use length contraction for such an argument. Length contraction in its normal form applies to objects moving at constant velocity. Going away from that you need to look into what has already been mentioned in this thread: Bell's spaceship paradox and Born rigidity.

 

[Arend]

The problem is that you cannot have the acceleration of two ends of an object starting simultaneously in two different frames of reference. SR is not just length contraction and time dilation - there is also the relativity of simultaneity.

Let's take an object of length $L$ at rest in some reference frame and consider two cases.

1) The object accelerates such that, in the original rest frame, both ends of the rod always have the same velocity. In this frame there is no length contraction, as the two ends remain a distance $L$ apart at all times.

Note that in this case, the object must be internally stretching. And, if the acceleration stops at some time, then in the new rest frame of the moving object it will have been stretched. In the rest frame of the rod, therefore, the front of the object must have been accelerating slightly more than the rear. It might be an good exercise to work this out fully.

2) If the object accelerates such that it retains it original length, then the acceleration of the two ends is not equivalent in the original rest frame. The rear of the object must have slightly more acceleration than the front in the original rest frame. So, it is not a rigid object moving at constant velocity. So, we do not have simple length contraction. Again, it would be a good exercise to work out the details fully.

Thanks PeroK. The supposition is an object in low velocity, so all parts are at the "same frame". In theory, you can have acceleration that meets the evaluation without approaching the light velocity. Applying the length contraction expression for low velocity objects, would result in an acceleration limit. I guess that perhaps, I am not allowed to simply derive the relativist expression in time.

 

[PeroK]

Here's an idea for analysing this problem.

1) In the original rest frame both ends accelerate simultaneously at some constant acceleration $a$ until they reach a velocity $v$. Then, consider a frame of reference moving at velocity $v$ relative to the original rest frame. In this frame, the object begins with velocity $-v$ and both ends decelerate to rest. However, using the Lorentz Transformation (or the leading clocks lag rule) we see that the front of the object started accelerating first in this frame.

2) Again consider a reference frame moving with the final velocity of the object. The object must have started length contracted in this frame. But, as it ended up with its rest length, it must have been measured to stretch in this frame. This time, the front of the object must have started and ended accelerating first.

In any case, it's ironic that in neither frame do we measure increasing length contraction of the object! In the first case, the object remains a fixed length. And, in the second, it starts length contracted and stretches as it decelerates. Note that the rate at which the object can stretch is limited by the speed of light, but this is compatible with unlimited acceleration - although the details depend on how you specifiy the acceleration profile.

 

[PeroK]

Thanks PeroK. The supposition is an object in low velocity, so all parts are at the "same frame". In theory, you can have acceleration that meets the evaluation without approaching the light velocity. Applying the length contraction expression for low velocity objects, would result in an acceleration limit. I guess that perhaps, I am not allowed to simply derive the relativist expression in time.

If we consider a single particle, there is no limit to its acceleration. You can see this by considering an electric field of arbitrarily large strength acting on a charged particle. Before you go any further, you must see that, by this simple example of a charge particle in an electric field, relativity specifies no maximum acceleration.

Therefore, your conclusion is clearly wrong. The question is why is it wrong? I've provided some analysis above. In a nutshell, it's the relativity of simultaneity - as it often is. It's a common mistake to forget about the relativity of simulateity and find a paradox in SR by using only length contraction and/or time dilation.

 

[Dale]

As was mentioned above, the length contraction formula assumes an inertially moving object. It doesn’t apply for accelerating objects.

In general, if a formula is derived using some assumption, then it cannot be used in a situation where that assumption is violated. At least, not without some careful justification (typically done by calculating the magnitude of the error introduced by using the assumption)

 

[Ibix]

Hi Ibix, fixed the attachments. Thank you for your comments. The point is that applying the length contraction expression for low velocity objects, would result in an acceleration limit.

For the length contraction formula to apply to an accelerating object you would have to assume a low enough acceleration that transient deformations were negligible and the velocity of every point on the object was nearly equal along its length. This is a fundamental issue in relativity, not dismissable with the "it's just an engineering problem" handwave you could make with Newton. So you're assuming a low acceleration and finding that you can't have a high acceleration with that assumption.

You really need to look up Bell's spaceships, as I guessed.

 

[berkeman]

a deep thank you for your patience with my amateur views.

(Thread prefix changed A-->I)

 

[Dale]

The relativity of simultaneity is the big issue with using the inertial length contraction formula for an accelerating objects. In the derivation you assume that the distance between the endpoints does not depend on time. That is only true in both frames for an inertial object.

 

[A.T.]

... after doing the time derivative of the relativistic length contraction, and assuming that the maximum length change in time would be the light velocity.

Relativistic length contraction relates the lengths of an object in different frames, not at different times during acceleration. You can accelerate an object without any change of its length as measured in its initial rest-frame (if the material is stretchable enough).

See the setup of Bell's spaceship paradox:
https://en.wikipedia.org/wiki/Bell's_spaceship_paradox

 

[Ibix]

One other point is that not everything that has dimensions $\mathrm{LT^{-1}}$ is a velocity. The limit to a rate of change of length is $2c$, not $c$, since you can hammer both ends of an object towards each other at just less than $c$ in opposite directions.

Note that this would cause the object to behave non-rigidly.

 

[Herman Trivilino]

TL;DR Summary: The relativistic length contraction expression should result in acceleration limitattion.

Doing something wrong?

Yes. The length contraction formula is derived under the condition that $v$ is strictly less than $c$. Thus you cannot make that substitution where you set $v=c$ in that formula.

 

[Arend]

For the length contraction formula to apply to an accelerating object you would have to assume a low enough acceleration that transient deformations were negligible and the velocity of every point on the object was nearly equal along its length. This is a fundamental issue in relativity, not dismissable with the "it's just an engineering problem" handwave you could make with Newton. So you're assuming a low acceleration and finding that you can't have a high acceleration with that assumption.

You really need to look up Bell's spaceships, as I guessed.

Hi Ibix, due to the relativistic acceleration behavior (or Bell's assumptions), could we assume that any object for getting accelerated shall suffer atomic deformation? In consequence, for example, from rest, will 1m 1kg of iron require lower energy than 1m 1kg of diamond to achieve the same velocity?

 

[Ibix]

Hi Ibix, due to the relativistic acceleration behavior (or Bell's assumptions), could we assume that any object for getting accelerated shall suffer atomic deformation? In consequence, for example, from rest, will 1m 1kg of iron require lower energy than 1m 1kg of diamond to achieve the same velocity?

I don't know what you mean by atomic deformation or 1m 1kg.

However, the only factor in the energy requirement to accelerate something to a given speed is the mass. The material does not matter unless you accelerate it hard enough to cause permanent damage (like shattering a plate against a wall) in which case some of the energy goes into doing the damage and the material will matter. Length contraction is not a deformation in that sense.

 

[Arend]

I don't know what you mean by atomic deformation or 1m 1kg.

However, the only factor in the energy requirement to accelerate something to a given speed is the mass. The material does not matter unless you accelerate it hard enough to cause permanent damage (like shattering a plate against a wall) in which case some of the energy goes into doing the damage and the material will matter. Length contraction is not a deformation in that sense.

Sorry the direct approach. Following Bell's observations, the relativistic movement between parts of the same object (or different as the illustrated rockets example connected by wire), will create an internal force or deformation that is not only related to the Newtonian forces. So the supposition is that materials with more flexible atomic structures experience "easier" relativistic acceleration with less energy expenditure. The 1m (length) 1kg diamond object will need more relativistic kinetic energy (besides the additional lattice deformation) than the iron object.

 

[Ibix]

Sorry the direct approach. Following Bell's observations, the relativistic movement between parts of the same object (or different as the illustrated rockets example connected by wire), will create an internal force or deformation that is not only related to the Newtonian forces. So the supposition is that materials with more flexible atomic structures experience "easier" relativistic acceleration with less energy expenditure. The 1m (length) 1kg diamond object will need more relativistic kinetic energy (besides the additional lattice deformation) than the iron object.

Bell's spaceships stress the string holding them together, but this is precisely because they don't follow the natural acceleration profile for an object of their length. In the case that you are increasing the stress in something then how much energy it takes to produce a certain stretch does depend on the material, yes.

But they could accelerate in a way that doesn't produce stress in the string (or only transients). That way, the material is unimportant - only the mass matters. The condition for this is that if the back rocket accelerates at $a$ and the front rocket is distance $L$ in front, it needs to accelerate at $ac^2/(c^2+aL)$ (if I did the algebra correctly in my head), and they need to stop accelerating simultaneously according to their final rest frame, not their initial one.

So the point here is that the material only matters if you stress it. Note that if Bell's rockets turn off their engines before the string snaps, the stress in the string will cause them to start oscillating about their center of gravity - it's this harmonic motion that gets the "extra" energy that's needed over the better planned acceleration. Note that there's nothing particular to relativity here except for the details of the best acceleration plan. If you stress a string in non-relativistic physics you'll get oscillation when you let it go too.

 

[PeroK]

Sorry the direct approach. Following Bell's observations, the relativistic movement between parts of the same object (or different as the illustrated rockets example connected by wire), will create an internal force or deformation that is not only related to the Newtonian forces. So the supposition is that materials with more flexible atomic structures experience "easier" relativistic acceleration with less energy expenditure. The 1m (length) 1kg diamond object will need more relativistic kinetic energy (besides the additional lattice deformation) than the iron object.

You need to be careful when analysing the Bell's spaceship as it can lead you away from an understanding of SR. In general, that is what paradoxes do. They are deliberately constructed to confuse. Paradoxes are better tackled once you have mastered the subject.

Length contraction is a coordinate effect. It's not a physical contraction that stresses the material.

 

[Arend]

Very interesting. So, for the case where it is applied a homogeneous acceleration to the object, the relativistic kinetic energy expenditure will be material dependent?

 

[PeroK]

So, for the case where it is applied a homogeneous acceleration to the object, the relativistic kinetic energy expenditure will be material dependent?

Homogeneous according to which frame of reference?

 

[Arend]

Homogeneous according to which frame of reference?

observer

 

[A.T.]

You need to be careful when analysing the Bell's spaceship as it can lead you away from an understanding of SR. In general, that is what paradoxes do. They are deliberately constructed to confuse. Paradoxes are better tackled once you have mastered the subject.

The Bell's spaceship scenario is only paradoxical, if you have already aquired a wrong understanding of SR and length contraction. The misunderstanding, that length contraction relates the lengths at different times during acceleration, rather than in different frames, is very common and leads to confusions like the OP has here. Bell's spaceship scenario is not constructed to confuse, but to chalange and dismantle this misunderstanding. If you introduce it early, along with lenght contraction, such misunderstanding might be avoided in the frist place.

 

[PeroK]

observer

In that case, the simplest statement is that the object must be physically stretched as it accelerates. Consider that the object consists of a number of particles in a line, initially a certain distance apart from each other. If the acceleration is homogeneous in the original rest frame, then the particles must get further apart in, to be precise, an instantaneously comoving inertial frame. And, in the final rest frame of the object once the acceleration phase has ended.

 

[Arend]

The Bell's spaceship scenario is only paradoxical, if you have already aquired a wrong understanding of SR and length contraction. The misunderstanding, that length contraction relates the lengths at different times during acceleration, rather than in different frames, is very common and leads to confusions like the OP has here. Bell's spaceship scenario is not constructed to confuse, but to chalange and dismantle this misunderstanding. If you introduce it early, along with lenght contraction, such misunderstanding might be avoided in the frist place.

Now I can see, I should go deeper before going into paradoxical ideas. But got (wrong) perception due to some older papers (including Bell) that claimed the stress condition under relativistic acceleration. However, Minkowski pointed out more than a hundred years ago that relativistic length contraction could not be understood as some kind of deformation effect, it is just apparent, referential frame related. Thanks

 

[A.T.]

But got (wrong) perception due to some older papers (including Bell) that claimed the stress condition under relativistic acceleration.

The streess will be there, if you accelerate something, while preventing it from contracting. Length contraction means, that if an accelerating object maintained its length in its intial inertial rest frame, then it must have been streched in its rest frame. And that imples stress.

However, Minkowski pointed out more than a hundred years ago that relativistic length contraction could not be understood as some kind of deformation effect, it is just apparent, referential frame related. Thanks

Length contraction by itself doesn't imply deformational stresses. But in combination with additional boundary conditions (like: maintain constant length in a certain frame) such stresses might be implied.

 

[Arend]

Mean stress due to direct relativistic effect. For sure any mass under acceleration gets stressed due to the applied force. No stress or deformation forces due to the Lorentz contraction.

 

[A.T.]

For sure any mass under acceleration gets stressed due to the applied force.

In a thought experiment you can distrbute those external forces to minimize any stresses from them.

No stress or deformation forces due to the Lorentz contraction.

No stress due to Lorentz contraction alone. But in combination with the boundary condition of constant length in the intial inertial rest frame, you will have stresses, no matter how you distrbute the external accelerating froces.

 

[Nugatory]

@Arend You may want to read some about "Born Rigid Motion"