The energy that depends on the observer

Two lumps of clay collide and stick. Six observers watch, drifting past one another at various steady speeds, and each computes the kinetic energy before and after. No two of them get the same pair of numbers. The largest total is five times the smallest. Every one of them is right, and none of them is measuring badly.

That gap is the same for all of them, to the last digit a double-precision computation carries. It is worth being clear how surprising that ought to be. Kinetic energy is quadratic in velocity, and adding a constant to a velocity does not add a constant to its square. Yet the difference between two such quantities, taken before and after an event, comes out independent of the constant. Something is cancelling, and what is cancelling is momentum.

The collision, in one frame #

Start with the ordinary account. Two bodies approach, interact, and leave; momentum is conserved because nothing outside pushed on the pair; energy is not conserved because the interaction was inelastic and some of it went into heating the clay.

Now watch the same event from a platform moving at $w$. Every velocity becomes $v - w$, the momentum becomes $p - Mw$ with $M$ the total mass, and the kinetic energy becomes

$$K' = \tfrac12\sum m_i (v_i - w)^2 = K - w\,p + \tfrac12 M w^2 .$$

Three terms. The first is the original energy. The third depends only on the total mass and the observer, so it is the same before and after the collision and cancels out of any difference. The second is $w$ times the total momentum — and momentum is conserved, so that term is also the same before and after, and it cancels too.

What is left is that $\Delta K' = \Delta K$ for every $w$. The energy destroyed in a collision is an invariant, and the reason it is an invariant is that momentum is conserved. Two conservation laws that are usually taught as separate facts are, in this one respect, the same fact.

The energy nobody can get below #

The middle term of that expression, $-wp$, is linear in the observer’s speed and the last is quadratic, so the total energy as a function of $w$ is a parabola with a minimum. That minimum is at $w = p/M$, which is the velocity of the centre of mass.

That decomposition has a name, König’s theorem, and it says the total kinetic energy of any system splits exactly into two pieces: the energy of the total mass moving at the centre-of-mass velocity, plus the energy of everything’s motion relative to the centre of mass. The first piece is entirely a matter of who is watching. The second is not, and it is the piece a collision can destroy.

Where a rolling body’s energy sits splits the same way: part in the motion of the centre, part in the rotation about it. That is König’s theorem again, and it is worth noticing that the second part is the one every observer agrees about. The internal energy of a system — rotation, vibration, heat — is frame-independent, and the part that moves with the observer is the part attached to the centre of mass.

This is why a collision experiment is set up, wherever possible, in the centre-of-momentum frame: not for elegance but because that frame contains no bookkeeping the physics did not put there. It is also why the centre of mass is the thing that keeps moving when everything else is thrown about — it carries the part of the motion nothing internal can touch.

What work does, in two frames #

The obvious objection is that the work–energy theorem must break. Work is force times distance; the distance a body moves depends on who is watching; so the work done depends on the frame, while the force does not. If the theorem holds in one frame it ought to fail in another.

It does not fail, and the reason it does not is the same cancellation. Consider a constant force $F$ acting for a time $t$ on a body of mass $m$. In the original frame the body starts at $v_0$ and covers $v_0 t + \tfrac12 a t^2$. In a frame moving at $w$ it starts at $v_0 - w$ and covers $(v_0 - w)t + \tfrac12 a t^2$ — a distance smaller by exactly $wt$. So the work is smaller by $F w t$, which is $w$ times the impulse, which is $w$ times the momentum change. And the kinetic energy change is smaller by exactly the same amount, from the middle term of the transformation above. Both sides shift together.

Every force in a free-body diagram is the same for every observer moving steadily. The displacements are not, and therefore the works are not — so two observers agree about the forces and disagree about the work done by each one, while agreeing about the total change in kinetic energy. That is not a paradox; it is what it means for the work–energy theorem to hold in every inertial frame at once.

The theorem is therefore frame-independent in the only sense that matters: it is true in every frame, while the numbers in it are true only of one. That is a stronger and stranger statement than “energy is conserved”, and it is what makes energy usable at all.

Where the fuel goes #

The awkward case, and the one that produces most of the confusion, is a car accelerating. Take it from rest to 10 m/s and its kinetic energy rises by $\tfrac12 m (10)^2$. Take it from 10 to 20 and the rise is $\tfrac12 m (400 - 100)$, three times as much — for the same force applied for the same time, and therefore, apparently, for the same fuel.

The apparent paradox has two halves and both are worth separating.

The first half: the fuel is not the same. A car that maintains a fixed force while going faster is delivering more power, and over the same interval it burns more. Force times distance is what the engine has to supply, and the distance in the second case is three times greater. There is no discrepancy here at all; the confusion comes from holding the wrong thing fixed.

The second half is the real one. Watch the same car from a van already travelling at 10 m/s. In the van’s frame, the first stage takes the car from $-10$ to $0$ and its kinetic energy falls to zero; the second takes it from 0 to 10 and the energy rises. The van’s occupants and the roadside observer disagree about which stage was expensive. They cannot both be describing what the engine did.

Within one frame the bookkeeping is exact and complete. A pendulum swaps kinetic for potential energy and the sum stays put, and nobody needs to mention observers at all — which is why the frame-dependence is so easy to miss. It only appears when the camera moves, and almost every elementary treatment keeps the camera still.

They are not both describing what the engine did, and the resolution is that neither of them was. The engine acts on the road as well as on the car, and the road is a body with an enormous mass. In the roadside frame the road does not move, so no work is done on it and the whole of the engine’s output goes into the car. In the van’s frame the road is moving at 10 m/s backwards, the friction force from the tyres acts on it, and work is being done on the road at a rate that exactly makes up the difference. Every frame’s books balance; what differs is which entry the energy is written against.

The frame in which the road is at rest is not privileged by mechanics. It is privileged by the fact that the road is where the heat ends up, and where the heat ends up is a question about a huge number of degrees of freedom rather than about a coordinate choice. That is the sense in which dissipation picks out a frame when the underlying laws do not.

Ordered kinetic energy is one arrangement and heat is an enormous number of them, which is the smallest honest statement of why the two are not interchangeable. A count of arrangements is the same for every observer — a count cannot depend on who is counting — so the entropy of a system is frame-independent even where its energy is not, and that asymmetry is what makes “where the fuel goes” a well-posed question with a frame-independent answer.

The point is worth stating plainly because it is easy to mistake for a loophole. Nothing in the transformation above cares which body is heavy. What the mass of the road buys is that its velocity change is unmeasurable while its momentum change is not — so the momentum bookkeeping needs it and the energy bookkeeping, in the frame where it is at rest, does not. Choose any other frame and the road re-enters the energy accounts immediately, carrying whatever is needed. A body large enough to absorb momentum without visibly moving is exactly a body whose energy contribution is frame-dependent all the way down, and the road, the Earth and the laboratory bench are all of them that body.

The third conservation law #

The observation that momentum conservation is what makes the energy loss invariant has a tidier statement than the algebra suggests, and it comes from asking which symmetry each conservation law belongs to.

Energy is conserved because the laws do not change with time; momentum because they do not change with position; angular momentum because they do not change with orientation. Those three are familiar. There is a fourth symmetry in the same family and it is the one this essay has been using: the laws do not change when the whole description is set in uniform motion.

Its conserved quantity is less famous and is easy to write down. It is the total mass times the position of the centre of mass, minus the total momentum times the time — a quantity whose constancy says exactly that the centre of mass moves in a straight line at a steady speed. That is usually presented as a consequence of momentum conservation, and it is a conservation law in its own right, belonging to boosts as momentum belongs to translations.

Once that is in view, the essay’s central cancellation stops being an accident. The energy an observer measures depends on their velocity because energy and momentum are components of one object that boosts mix; what does not depend on it is whatever the mixing leaves alone. In Galilean mechanics that is the energy in the centre-of-momentum frame; in relativistic mechanics it is the invariant mass, and the two are the same statement with the same origin.

Which is why the collection of results in this essay hang together so tightly. König’s theorem, the invariance of the dissipated energy, the survival of the work–energy theorem and the existence of a frame that minimises the total are four faces of one symmetry, and none of them has to be checked separately once the symmetry is stated.

The burn that is worth more at speed #

The car’s paradox has a version that is not a paradox at all but a technique, and it is used on every mission beyond the inner solar system.

A rocket firing its engine adds a fixed change of speed for a given expenditure of propellant, because what the engine does is throw mass backwards at a fixed exhaust speed. But the energy the vehicle gains from a given change of speed is not fixed: going from $v$ to $v + \Delta v$ raises the kinetic energy by $mv\Delta v$ plus a small term, so the same burn is worth far more when the vehicle is already moving fast.

Which means a spacecraft should burn at the bottom of a gravity well, where it is moving fastest, rather than at the top. Dropping toward a planet, firing at closest approach, and climbing out again delivers more energy for the same propellant than the same burn made far away — sometimes by a large factor, and it is what makes several outer-planet missions possible at all.

The essay’s question applies immediately: an observer moving with the vehicle before the burn sees it start at rest, so the burn appears to buy the small term only. Where did the rest go?

Into the exhaust, and the accounting closes exactly. The propellant leaves at a fixed speed relative to the vehicle, so in the ground frame a fast vehicle’s exhaust ends up nearly at rest — carrying away almost no kinetic energy — while a slow vehicle’s exhaust is thrown backwards at nearly the full exhaust speed and takes a great deal. The chemical energy released is the same in both cases and in every frame; what differs is how it is divided between the vehicle and what it threw away.

So the technique is not free energy and is not a frame trick. It is a redistribution, arranged so that the part that leaves takes as little as possible, and the arrangement is available only when the vehicle is already moving fast in the frame the propellant is being left behind in.

Why thermodynamics never asks #

There is a place where all of this is quietly assumed and never discussed, and noticing it is a good test of whether the argument has landed.

Thermodynamics deals in a quantity called the internal energy, and no thermodynamics textbook ever says which frame it is measured in. It does not need to, because internal energy is defined as the energy in the frame where the body as a whole is at rest — which is König’s decomposition with the first term discarded by fiat.

That is why a gas in a moving railway carriage has the same temperature as one at rest, why heating a body does not change its momentum, and why the first law can be written with no velocity in it. All three are consequences of having thrown away the frame-dependent piece before starting.

The convention is so well hidden that its failures are surprising when they arrive. A gas with a bulk flow in it has kinetic energy that is not internal, and separating the two is the first thing any fluid calculation has to do — which is why the energy equation in fluid mechanics carries both a temperature and a velocity and why the sum of them is what is conserved. And a body moving fast enough for relativity to matter has an internal energy that contributes to its mass, which is the point at which the discarded term and the kept one stop being separable at all.

The rifle, which makes it unarguable #

The clearest version of the argument uses a system with two comparable masses and no road at all. A rifle fires a bullet. Momentum is zero before and zero after, so the bullet and the rifle carry equal and opposite momenta and their kinetic energies are in the ratio of the inverse masses: the light bullet takes nearly everything.

Now watch from a frame moving with the bullet’s final velocity. In that frame the bullet ends at rest and the rifle recoils at a much greater speed than before, and the rifle now carries almost all the kinetic energy. The propellant released the same chemical energy in both accounts. What it produced, in the two tellings, went to opposite ends of the apparatus.

The rifle makes it unarguable, and a potential landscape shows why. The height of the total-energy line is set by an arbitrary choice of where the potential’s zero sits, and no measurable quantity depends on that choice — only differences matter. Kinetic energy is in the same position with respect to the choice of frame: the value is a convention, and the differences are physics.

Both quantities in mechanics that are called energy therefore have an arbitrary constant in them. Potential energy has one because the zero of a potential can be put anywhere. Kinetic energy has one because the zero of velocity can be put anywhere. Neither arbitrariness leaks into anything measurable, which is exactly what an arbitrary constant is allowed to do.

Where the model stops #

A frame here is an inertial frame. Everything above uses a steady $w$, and the cancellation depends on it: an accelerating observer sees forces that do work and finds the energy books balanced only after the fictitious terms are entered. Those terms are not a repair of a broken law; they are what the transformation produces when it is done properly.

The total momentum has to be conserved for the loss to be invariant. If an external force acts during the collision — a wall, a rail, a hand — then $p$ changes, the $-wp$ term no longer cancels, and different observers really do disagree about how much energy was destroyed. Every statement here is about an isolated pair, and an experiment that clamps one of the bodies is not one.

And the invariance is Galilean, which is to say approximate. At speeds where the Lorentz transformation matters the whole decomposition changes shape: what survives a change of frame is not the energy difference but the invariant built from energy and momentum together, and the mass of a system turns out to be exactly the quantity this essay has been calling the internal energy.

Everything in this essay is one line of a spacetime diagram. Two observers on one picture — upright axes for the first, scissored for the second — disagree about how much of a displacement is “time” and how much is “space”, and energy and momentum are components of a single object resolved along those axes. Frame-dependence of energy is component-dependence of a vector, and nothing more mysterious than that.

The low-speed end of the relativistic energy curve is the $\tfrac12mv^2$ of everything above, sitting on top of a constant Newtonian mechanics had no reason to notice. That constant is the rest energy, it is the same in every frame, and it is the invariant this whole essay has been circling — the part of the energy that does not depend on the observer.

What the pictures cannot show #

None of these figures can show an observer. Every one of them is drawn in some frame, on paper that is not moving, and the frame-dependence has to be represented as an axis or as a set of cases rather than as a picture — which is why the hero figure has the observer’s own speed along the bottom, a coordinate no drawing can ever contain.

Nor can any figure show which frame is the right one, because there is not one. The temptation to draw the “true” energies and then the distorted ones is strong and there is nothing to base it on: the figures here draw six sets of numbers and refuse to mark any of them.

Where this ladder goes next #

The rung below, the hill that gives it back, establishes energy as a quantity worth having: a scalar that can be traded between forms and counted at the ends of a process without following the middle. This rung is the price of that convenience — the number counted is a number about the counter, and only what changes is about the world.

The habit worth carrying is a test that costs nothing and settles a surprising number of arguments. Ask what happens to a quantity when the observer starts moving. A quantity that changes is a bookkeeping device; a quantity that does not is a candidate for being a property of something. Momentum fails the test and is a bookkeeping device. Kinetic energy fails it. The energy dissipated passes, the mass of a system passes, and — one collection over — the interval between two events passes, which is why relativity is built on that and not on time.

What is left on this ladder is where the dissipated energy went, which is a question about the number of ways a system can hold it, and belongs to the counting rather than to mechanics. The next rung against this anchor will be about power rather than energy — the rate at which a machine can do the trading, which is the quantity every engine is actually specified by and the one the ceiling on every engine constrains from the other side.