# How is energy conserved?

From what I’ve heard, there is no absolute grid you can put on the universe, and therefore you can only ever talk about relative motion.

I’ve also heard that energy is conserved and constant no matter your location or velocity.

But how does this work with kinetic energy? Surely if I start walking in a certain direction, all the galaxies in that direction have decelerated, and those behind me will have accelerated by the same amount. But since the equation for kinetic energy is $E = (1/2)mv^2$, this means that the galaxies which have, from my perspective, accelerated, have gained more kinetic energy than those which have, from my perspective, decelerated. How does this work? I’m assuming that the energy changes that happen when I walk are insignificant compared to the extra kinetic energy gained by billions of galaxies.

• Be careful with the word "decelerate". I find that it causes more confusion that it's worth, and I avoid using it. Both the galaxies in front of you and those behind you are speeding up. – garyp Apr 5 '18 at 16:56
• Different inertial observers see different energies. But the energy observed by a fixed inertial observer never changes. When you changes from one inertial frame to another inertial frame, energy does change. – velut luna Apr 5 '18 at 16:57
• But how do you define an inertial observer? We cannot even define it as being an observer who isn’t accelerating as all galaxies are accelerating away from each other. – Arkleseisure Apr 5 '18 at 17:01
• If you want to involve the expansion of the universe, then you have to involve GR and it gets more messy. Your question doesn't really touch on that, so I gave a Galilean answer. The statement about "energy is conserved" is no longer simple under GR either. physics.stackexchange.com/questions/35431/… – BowlOfRed Apr 5 '18 at 17:12
• Possible duplicate of Kinetic energy with respect to different reference frames – sammy gerbil Apr 5 '18 at 18:56

There is no single preferred frame for examining the universe, but that doesn't mean all frames are identical. In fact, for many forms of analysis, we require an inertial frame.

The fact that you start walking does not mean that every other object in the universe has accelerated. It means that you are examining them from another frame of reference or you are using a frame that is itself accelerating (non-inertial).

Energy is not conserved when translating between different frames of reference. To keep things sane, you need to pick one (inertial) frame of reference and stick to it.

You have two obvious ones in your scenario. One frame in which you are at rest before you begin walking, and a frame in which you are at rest after you begin walking.

In either frame, the only object that we consider to change speed is you, so your KE changes before and after $t=0$, while the other objects do not.

Let's tackle this problem from the perspective of Newtonian mechanics since that will be most illuminating to the confusion at hand. In Newtonian mechanics, there are special preferred reference frames, inertial reference frames, which you can define as a frame in which no pseudo forces arise - i.e. a frame in which Newton's first law applies. This problem you are seeing is then simply a confusion between the two different concepts of conservation and frame invariance. Energy is conserved but it is not frame invariant. Different inertial reference frames will measure different energies of objects, but a given inertial reference frame will always measure total energy to be constant. Hopefully that distinction is clear. Now, in this context, when you are accelerating you are changing from one inertial frame (not walking) to another (walking at constant pace - after you finished accelerating) and so the energy that you measure will NOT have remained the same since you have changed reference frames. You have simply shown that Energy is not frame invariant. During the acceleration phase, you are in a non-inertial reference frame so your analysis is not preferred in Newtonian Mechanics. You would have to introduce pseudo force(s) to explain why everything else is accelerating around you even though no apparent forces are acting on exterior objects (i.e. why is Newton's first law failing for you).

• I know this is a mostly unrelated area of physics but could dark energy therefore simply not be required from a different reference frame? – Arkleseisure Apr 5 '18 at 17:23
• What do you mean by "required"? The dark energy is a funky concept to regard in the same terms as "regular" energy. Evidence of its existence is based on cosmological observations and trying to fit those observations to current models. There's no way (in our current model of the universe) to transform to a frame where there is no dark energy . – enumaris Apr 5 '18 at 17:35