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Thought experiment: I am starting to walk on a corridor. From the corridor's perspective I have a certain increase of kinetic energy. From my perspective the whole universe is gaining kinetic energy because it has a speed relative to me. The kinetic energy increase from the corridor's point of view is definitely not equal to the kinetic energy increase of the universe from my perspective. What am I doing wrong?

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BinaryBurst
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    You are concerned that this doesn't comply with energy conservation law, am I correct? If not, why do You think that these two should be equal? – Wojciech Mar 26 '14 at 14:01
  • Yup conservation of energy – BinaryBurst Mar 26 '14 at 14:02
  • "The law of conservation of energy states that the total energy of an isolated system cannot change". Is Your system the same in both cases? – Wojciech Mar 26 '14 at 14:06
  • Yes, my universe is a 10 trilion ton closed coridor – BinaryBurst Mar 26 '14 at 14:06
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    I see it that way: In the first case You are the system and the corridor is an observer, not part of the system. In the second case, the corridor is the system and You are an observer, not part of the system. So the conservation of energy doesn't apply here. – Wojciech Mar 26 '14 at 14:10
  • So I cannot measure kinetic energy while moving? And agree with somebody outside the coridor about what happened with the amount of energy? – BinaryBurst Mar 26 '14 at 14:14
  • I don't quite undestand what You're asking right now. But if You want to apply conservation of energy You need to investigate some form of changes of energy. I this case, You're simply comparing energies of different systems. – Wojciech Mar 26 '14 at 14:19
  • ok... First case: observer stationary relative to coridor, measures change in my kinetic energy. Second case: observer stationary relative to me, measures change in coridor's kinetic energy(way larger than the one from first case). Why aren't the changes equal since the observers measure the same system? – BinaryBurst Mar 26 '14 at 14:23
  • It doesn't matter, he still measures energy of different systems. You do not exchange energy with corridor, so Your presence does not affect the measurement. If the observer is stationary in relation to You, than You are needless. – Wojciech Mar 26 '14 at 14:25
  • If sth is still unclear to You, let me know. – Wojciech Mar 26 '14 at 14:49
  • Check this article on Mach's Principle. It considers a rotational version of the same question. http://en.wikipedia.org/wiki/Mach%27s_principle There are variants of Mach's Principle. Note Mach2. http://en.wikipedia.org/wiki/Mach_principle – mmesser314 Mar 26 '14 at 14:51

2 Answers2

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There are two distinct points to make, and both are related to the implicit assumption of Galilean invariance that you're making. Galilean invariance is the idea that all inertial frames are equally valid so, for example, you might stand by a train track watching the train and regard yourself as stationary and the train as moving, but a passenger on the train would be equally justified in regarding themselves as stationary and you as moving. In this case the choice is between you moving and the corridor stationary and the corridor moving and you stationary.

The first point is that kinetic energy is not an invariant under Galilean transformations. This is obviously so since if a fly is moving towards an elephant the total kinetic energy measured in the rest frame of the fly is (far) higher than the total kinetic energy measured in the rest frame of the elephant. However this does not violate conservation of energy, it just moves the reference point that we define as zero energy. No energy is appearing or disappearing.

The second point is that Galilean invariance applies only between inertial frames, i.e. frames moving at constant velocity, and introducing acceleration breaks the invariance. Velocity is relative, and you can't say what your velocity is or what the corridor's velocity is because it depends on the observer. However you can always say what your acceleration is because you can measure it using an accelerometer (like the one in your smartphone) without referring to any other object.

This matters because with acceleration you do have to worry about conservation of energy. When you begin walking you experience some force, and the work done on you is equal to force times distance moved. This work is equal to the increase in your kinetic energy. Likewise the corridor (and presumably the planet Earth) feels an equal and opposite force and it's kinetic energy must increase by the force times the distance the corridor moves.

You can choose any inertial frame to watch your progress down the corridor, but in all frames the force on you and the force on the corridor will be the same. It's straightforward to demonstrate the the work done is equal to the total kinetic energy change in all frames, but I won't do it here since this is dicussed in detail in the question Perspective and changes in kinetic energy.

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John Rennie
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Let us assume as you move (velocity $v$) you will see the relative kinetic energy of the Universe increases, therefore energy of $KE_0$ we measure is different to the kinetic energy in perspective of the Universe of us $KE_1$, and since the mass of universe is greater than you we can assume the kinetic energy will also be greater, therefore we will model this situation mathematically and say: $KE_0 > KE_1$ therefore observer will assume the energy conversations are broken. However now, each object in the universe will move away from other objects relatively at velocity $v$ as well therefore the relative kinetic energy between any 2 given coordinates will always remain $0$ in any & all frames of reference as the distance between them will never change but remain constant, so the 2 objects can never collide.

Using the above statement of

so the 2 objects can never collide

We can deduce that the $KE_0$ is not kinetic energy but should be considered more of a potential energy and since we must release this potential energy to break conservation of energy as we would create more energy and since using the statement of the 2 objects never colliding we can safely say the relative potential energy may never be harnessed by an means.

In other words, rather than looking at the relative kinetic energy we should consider it "relative potential kinetic energy". Finally, we can deduce that since we cannot directly create any energy using this "relative potential kinetic energy" the observer observers , we cannot have broken any laws of physics as no energy is created in any frames of reference.