# A mechanical system in special relativity

#### Methavix

hi all, i woul like to study a simple mechanical system with the special relativity.
a spring-mass-damper (viscous damping) parallel to the flight direction (we suppose to have the system attached to a relativistic spacecraft). i would like to study the case with an external force (constant) and without it.
we have a proper frame (attached to the spacecraft and the system) moving with a constant relativistic velocity V, and a coordinate frame at rest on the earth.

i would like to write the motion equations of the mechanical system in the proper frame and in the coordinate frame also in order to yield the spring constant and the damping factor in both of the frames.

someone said me to write the lagrangian in both of the frames (but i'm not able to do it correctly) and someone else said to write directly the motion equations in the proper frame and transform all quantities to the coordinate frame. but in this second way, how can i do to transform the sprign constant? and the damping factor?

besides, if te mechanical system is perpendicular to the flight direction?

in both of the cases, only V is relativistic, the mass motion (in the proper frame) is very slower.

thank you very much for your help
Luca

Related Special and General Relativity News on Phys.org

#### Methavix

thanks for the link, very interesting but I think that I can't use it for my problem...

#### Chris Hillman

Maybe I was being too subtle: in Greg's analysis, he is appealing to Hooke's law only in a quasi-static situation, not a dynamical situation (where the forces are associated with moving stuff). That is good, because Hooke's law invokes instantaneous response throughout the spring, which is incompatible with str. In your situation, OTH, my first thought is that when you try to write down an equation describing your damper, you will probably reach for the tried and true (damped harmonic motion), but you will also need to think about whether that even makes sense. Is the damper in your scenario supposed to be moving at non-relativistic velocities wrt the cabin of the spacecraft?

Actually, I think you need to clarify much more: at the moment, I don't understand what relativity has to do with any of this. Why would a Newtonian analysis be unsatisfactory in the scenario you have in mind? Why insist on treating the problem in a coordinate chart comoving with the Earth, rather than the spacecraft (which you said is not accelerating).

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#### lalbatros

Try to write the equations of motion of your system in the frame of the spacecraft.
Then convert that to any other inertial frame of reference.
Have a look at this result and decide if it looks like the original equations.
What are the changes you need to do to make it look like these original equation.

As a guess, the length of the spring and the relative velocity of the "piston" in the damper cylinder will be the key points.

#### Chris Hillman

Hi again, Methavix,

I didn't explain myself very clearly. Let me try again:

All other things being equal, as a way to master relativistic physics (if nothing else), it is very reasonable to systematically take everything you know from non-relativistic physics and try to figure out what the relativistic generalization would be. So I wasn't trying to suggest that your plan might be unwise for that reason.

Rather, I was trying to warn you of something which you probably couldn't expect, but which is widely appreciated by thoughtful students with more experience with this stuff. Namely: certain problems which seem very natural--- indeed, which are very natural--- turn out to involve (seemingly) every conceptual and technical issue which has not yet been satisfactorily resolved, which is why the literature on these problems is comparatively small and certainly much more confused than the literature on other problems which are more tractable with current knowledge.

I am not quite saying that it is impossible to treat elastic bodies in relativistic physics, or to treat "interesting" dynamical processes such as (say) an explosion which generates a relativistic shock, or the relavistic collision end to end of two elastic or plastic rods, just that this kind of problem turns out to be much trickier than most people realize. (In the confused literature dealing with such problems, a good deal of the confusion is due to physicists who ought to know better failing to recognize the subtleties or the multiplicity of issues they must deal with to come up with reliable results.) From my experience in reading the literature, I am concerned that you might get a lot of well meaning advice from people who might know a good deal about standard topics in relativistic physics, but almost certainly have not thought about the question you asked and who are therefore likely to vastly underestimate the potential difficulties. Thus, I fear that you might be responses suggesting various "easy approaches", which would probably be very misleading. So just be very careful, at least until you have a lot more experience in the field and know the literature on this kind of vexed topic much better.

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#### lalbatros

Chris,

I understand what you are saying to Methavix and I agree that "Newtonian problems" usually need some work to be translated consistently to special relaticity.

For example, a standard spring model as in the problem discussed now doesn't exist really in relativity. I mean that a force proportional to the elastic deformation of a spring, F=-kx, is not compatible with relativity since it would imply that this force propagates faster than light from one end to the other end of the spring. Of course it should not be so difficult to come with a full relativistic model. Cutting the spring in small pieces with non-relativistic relative velocities would do the job (local equations of motion for each element). But is there a need to go so far here?

I think that the question by Methavix is a simple homework involving lengths and velocities under Lorentz transformation, nothing more. As usual for elementary homeworks, there are implicit assumptions that are not explained because they are beyond the scope of the course or the exercice. Here, probably, the implicit assumption is that all the relative velocities in the spring-damper-mass system are much smaller than the speed of light. Then there is no need for complicated developments, just apply the Lorentz transformation, manipulate definitions, and get the grade.

Michel

#### Chris Hillman

I think that the question by Methavix is a simple homework involving lengths and velocities under Lorentz transformation, nothing more. As usual for elementary homeworks, there are implicit assumptions that are not explained because they are beyond the scope of the course or the exercice. Here, probably, the implicit assumption is that all the relative velocities in the spring-damper-mass system are much smaller than the speed of light.
You well could be right. I myself hinted at that in one of my comments above.

Methavix? Was this a homework problem? If so, did we guess correctly that you omitted the assumption that the damper is nonrelativistic (in the frame of the spacecraft)?

#### Methavix

lalbatros is right, in this problem the mass-spring system is non-relativistic, sorry for the oversight.
this is not a homework but a possible real problem for a relativistic spacecraft. if we have a relativistic spacecraft (uniform motion) and into the spacecraft there is a spring-mass system (and a spring-mass-damper system) we can study there everything. we can calculate the motion equation directly or through the lagrangian. it's easy. but if we want to "translate" these equation to a coordinate system what do we have to do? this is my problem, also to calculate the expression of the spring constant and damping factor in the coordinate system.

thank you very much

#### Chris Hillman

Well, if your relativistic spacecraft is not accelerating ("uniform motion"?), then this is indeed trivial. Now I am confused again about what you want to do.

#### Methavix

the spacecraft, in the first case i'm studying, is not accelerating, it is in uniform motion. can you help to write motion equations in coordinate system?

the second case i want to study is the hyperbolic motion for the spacecraft (i.e. constant proper accelaration).

thanks

#### Chris Hillman

Well, special relativity suffices in both examples.

The second case involves a Rindler observer.

Inside the cabin of the spacecraft, in the first case, nothing strange is observed (unless you are not accustomed to being in a state of freefall, that is). In the second case, you experience a constant acceleration in a certain direction (which might be more familiar to organisms who live in a small area on the surface of the Earth, which can be approximated as in the Einstein elevator).

#### Methavix

yes i know this, but my problem is to write the motion equations for the mass-spring-damper system both in the proper system (it is easy because they are the simple classical equations) and in a coordinate system (e.g. on the earth, supposing the earth at rest and inertial frame).

may you help me to write these?
thanks

#### pervect

Staff Emeritus
Are you familiar with 4-vectors, Methavix, or are you looking for an approach just using 3-vectors, i.e. x,y,z,t and x', y', z', t' coordinates?

#### Chris Hillman

Proper system?

yes i know this, but my problem is to write the motion equations for the mass-spring-damper system both in the proper system
You mean, a Cartesian chart comoving with your unaccelerated rocket? Vis a vis a Cartesian chart comoving with the Earth (or rather, I guess, with a nonrotating object other than the rocket)?

If so, if you can write down the motion in one chart, you can transform a la Lorentz to obtain the motion in the other chart, right?

In some other threads, I and pervect and others have discussed the fact that strictly speaking, the usual theory of simple harmonic motion (allowing for driving forces or impulsive blows) is incompatible with str since it postulates an instantaneous response throughout the spring to an impulsive blow. Depending upon your purposes, if the motion of the spring system in the chart comoving with the rocket is nonrelativistic, you can probably ignore this problem.

#### Methavix

to pervect: i'm not familiar with 4-vectors, i prefer 3-vectors.

to Chris Hillman: i mean a cartesian frame comoving with the spacecraft (where it is the meachanical system that it is nonrelativistic itself) and a cartesian frame at rest (i suppose that earth is at rest, or if you prefer a frame in the center of the solar system).
i can write the equations in the proper system (they are easy) but how can i do to transform them? i mean, i don't know how to transform the spring constant and the damping factor.

in the proper frame (i use ' sign) i write:

m'*(d^2/dt'^2)x'+b'*(d/dt')x'+k'*x'=0

where b' is the damping factor in the proper frame and k' is the spring constant in the proper frame.

and in the coordinate frame (it is at rest and the proper frame moves in respect to it with a constant velocity V, positive according to x).

thanks

#### Chris Hillman

I still don't see where this is going, but my suggestion was that you write out the equations of motion $x(t), \, y(t), \, z(t)$ of the endpoints of the spring in the chart comoving with the rocket, figure out how to reinterpret that in spacetime, and then transform to the chart comoving with your nonrotating Earth.

#### Methavix

the equation i wrote is the equation of motion of the endpoints of the spring in the chart comoving with the rocket, that is the equation of the classical physics.
i don't know how to do the lorentz transformation, this is my problem.

anyway i want to solve this problem just to see the equations of motion as seen by the earth :)

#### Methavix

i can solve the equations of motion in the proper system, but i continue to have the constant spring (or the frequency of the spring if you prefer) and i can't transform it.
i know how to tranform x, dx/dt and d^2x/dt^2 but i cannot do it for k (and for the damping factor if there is a damper).

#### Chris Hillman

To repeat: I suggested that you transform a typical solution, not the equations of motion.

#### pervect

Staff Emeritus
I've been meaning to give this thread some more attention, but I don't use the 3-vector approach very much, and I'm finding it awkward to formulate the problem properly with 3-vectors.

#### Methavix

To repeat: I suggested that you transform a typical solution, not the equations of motion.
for simplicity I suppose an undamped system. so I have:

m'*d2x'/dt'2+k'*x'=0

and the solution is

x'(t') = x'(0)*cos[sqrt(k'/m')*t']

then, i make this substitution:

x'(t') = (x(t)-V*t)*gamma

m' = m*gamma*[1-(u(t)*V)/c^2]

t' = gamma*[t-(x(t)*V)/c^2]

and I yield:

(x(t)-V*t)*gamma = x(0)*gamma*cos[sqrt(k'/(m*gamma*(1-(u(t)*V)/c^2)))*gamma*(t-(x(t)*V)/c^2)]

if i did right, what can i do to continue?
thanks

I've been meaning to give this thread some more attention, but I don't use the 3-vector approach very much, and I'm finding it awkward to formulate the problem properly with 3-vectors.
sorry but i'm not able to use 4-vector s. can you suggest the right way to do it in this problem? thank you

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Staff Emeritus