Principle of relativity and energy?

[FeynmanMH42]

I was recently having an IM conversation with my friend about the principle of relativity - I'd been reading up on special and general relativity all day. Special relativity is based upon the postulate that all uniform motion is relative, and general relativity extends this to non-uniform motion, and non-inertial reference frames. So how does this tie in with the role of energy in classical mechanics?

Me:
Okay...
You know how when you accelerate one way, you can say the whole universe accelerates the other way? Relative motion?
So how does this work for energy?
Doesn't it take energy to accelerate something? Force = mass x acceleration, and work is done when a force moves something by a distance, and energy is expended when work is done
so wouldn't there be a different amount of energy expended to accelerate a small mass (you) as opposed to the mass of the whole universe except you?

Friend:
That's a little like the twin paradox..
The resolution of which is the fact that it's you that's being forced to move, not the universe.

Me:
So what's all this about all reference frames being equivalent?
The Principle of Relativity... all reference frames are equally valid ways of looking at a situation.
To say you accelerate one way is equivalent to saying the universe accelerates the other way. Wouldn't saying you're being forced to move and not the Universe invalidate this?

My friend didn't know the answer to this, and we changed subject. I'm still 17 and though I'm interested in physics my knowledge of relativity still comes from popular science books, so I haven't had any formal teaching in it yet. I'm hoping there's an obvious flaw in my logic. Can someone help me and point it out? Thanks.

 

[Mentz114]

Hi FeynmanMH42,

special relativity is indeed about the equivalence of, and transformations between inertial frames. And the fact that uniform motion is relative and undetectable in the rest frame.

But accelerating frames are not inertial and acceleration is not relative. If you have one non-accelerating frame and another that has its rockets firing, it is clear to everyone which frame is accelerating. Observers in the accelerating frame can detect it ( as you could in an accelerating car). So acceleration ( and rotation) are absolute in that sense, not relative.

So if you are on an accelerating ship, it would not be correct to say that you are stationary and the universe is accelerating in the opposite sense.

M

 

[DrGreg]

FeynmanMH42, everything Mentz114 says is true in the context of Special Relativity, but that doesn't really answer your point about General Relativity.

I would clarify that all inertial observers agree whether an object has zero acceleration or not, but they disagree on the numerical value of the acceleration when it non-zero. However, they all know how to calculate the "proper acceleration", which is what the object actually "feels" itself, and is defined to be the acceleration as measured by a "co-moving" inertial observer traveling at the same speed as the object at that moment in time. Thus the proper acceleration is absolute, something all observers can agree upon.

When we consider non-inertial observers, the maths gets more complicated and different observers may disagree on whether an object is accelerating relative to the observer's frame or not ("coordinate acceleration"), but they nevertheless agree on what the proper acceleration is. It can also be shown that, roughly speaking, force relates to proper acceleration rather than coordinate acceleration.

 

[Dale]

Doesn't it take energy to accelerate something? Force = mass x acceleration, and work is done when a force moves something by a distance, and energy is expended when work is done
so wouldn't there be a different amount of energy expended to accelerate a small mass (you) as opposed to the mass of the whole universe except you?

Hi FeynmanMH42,

Mentz114 is correct, however there is an even more important issue here. Energy is relative in the same way that time is relative. This means that the value of energy is relative to the reference frame (or coordinate system) in which you are measuring it, and it is meaningless to talk about energy without specifying the reference frame. You therefore cannot compare energy measured in one frame to energy measured in another frame in any meaningful way.

So the fact that the energies are different in the two reference frames you described is not surprising nor important. In each reference frame you would find that energy is conserved, but the two reference frames would generally disagree about the details.

 

[FeynmanMH42]

Thanks for the input, but I've had another thought... what about momentum in inertial reference frames?

If I'm traveling at 10 metres per second one way, it's fair to say I could consider myself at rest and say the room is traveling at 10 metres per second the other way. Since me and the room have different masses, you could say that for different observers there are different momenta involved, and overall momentum will be conserved.

But if I collide with the wall and come to rest, then the change in momentum - the impulse - would be different depending on whether you take my reference frame or the wall's reference frame. From the wall's reference frame, a relatively small mass is colliding with it and therefore the change in momentum will be small, the force imparted will be small and the damage will be minimal. But from my reference frame I collide with a huge wall, with a huge mass, and the change in momentum will be large - and therefore by I = Ft (since it takes the same amount of time) the force will be much larger. How does this work? Damage isn't relative... I know that bringing me (or the wall) to rest with a collision is accelerated and not uniform motion, but if we were in uniform motion before the collision then surely this has an affect on our momenta during the collision?

In fact, if I'm traveling one way at ten metres per second, I could say that the whole Universe is traveling the other way at ten metres per second. Since the Universe could contain an infinite amount of matter this means from my reference frame the Universe has an infinite momentum? Sounds to me like something's gone wrong...

 

[JesseM]

But if I collide with the wall and come to rest, then the change in momentum - the impulse - would be different depending on whether you take my reference frame or the wall's reference frame. From the wall's reference frame, a relatively small mass is colliding with it and therefore the change in momentum will be small, the force imparted will be small and the damage will be minimal. But from my reference frame I collide with a huge wall, with a huge mass, and the change in momentum will be large

Why do you think the change in momentum will be large? In this frame just as the other frame, your velocity changes significantly but the wall's hardly changes at all, so even though its momentum is large its change in momentum is not very large.

edit: I assumed you were talking about the change in the wall's momentum here, obviously the change in momentum of the combined system of you plus the wall will be zero.

 

[dkgolfer16]

From the wall's reference frame, a relatively small mass is colliding with it and therefore the change in momentum will be small, the force imparted will be small and the damage will be minimal. But from my reference frame I collide with a huge wall, with a huge mass, and the change in momentum will be large - and therefore by I = Ft (since it takes the same amount of time) the force will be much larger.

Don't forget about Newton's Third Law.

 

[Mentz114]

You are only colliding with the wall, and whatever it's attached to ( the Earth ?). Conservation of momentum will hold and the momentum_wall + momentum_person will be the same before and after the collision, from any frame of reference.

You should look up 'conservation of momentum' in this forum or Wiki. It's an interesting subject and fairly important.

I'm sure someone will give you a longer answer before long.

[already !]

M

 

[Dale]

Thanks for the input, but I've had another thought... what about momentum in inertial reference frames?

You have actually stumbled on something very useful. As you have discovered, both energy and momentum are frame-variant quantities. In SR energy and momenum are combined in the same way that time and space are combined into spacetime. The combination of energy and momentum is called the http://en.wikipedia.org/wiki/Four-momentum" , and it is one of my favorite concepts in SR.

In one neat little package you get conservation of mass, conservation of energy, and conservation of momentum. You also get a nice convenient way to Lorentz transform energy and momentum between different reference frames. I highly recommend you look at it.

 

[FeynmanMH42]

In this frame just as the other frame, your velocity changes significantly but the wall's hardly changes at all

I don't get this bit. True, you can say that from the wall's frame of reference my velocity changes from 10 metres per second to zero, but can't I say the same for the wall? I can consider myself at rest and the wall's velocity changes from 10 metres per second to zero?

 

[granpa]

acceleration is not relative.

 

[JesseM]

I don't get this bit. True, you can say that from the wall's frame of reference my velocity changes from 10 metres per second to zero, but can't I say the same for the wall? I can consider myself at rest and the wall's velocity changes from 10 metres per second to zero?

Not if you stick to inertial frames (the only frames where momentum is guaranteed to be conserved), no. In the inertial frame where you were at rest prior to the collision and the wall was moving, after the collision the wall's velocity will have changed only slightly in this frame, while your velocity will have increased to nearly twice the wall's velocity (assuming it's much more massive than you). A frame where your velocity is zero both before and after the collision is not an inertial one, so the usual laws of mechanics (both in Newtonian mechanics and in relativity) don't apply.