Q: How can something have different amounts of energy from different points of view?

The original question was: … in a scenario with two cars driving towards each other, the system could be measured externally to have an energy equal to the sum of the kinetic energy in the two cars. However, if you are in one of those cars, you would see the other car moving towards you at twice the speed you are traveling, and that you are not moving at all. If that is the case, then calculating the energy in the system means instead of summing the energy of two cars moving, you have one car moving at twice the speed, which means four times the energy of one car and twice the energy of the original system.  How is it that from one perspective, a system can have twice as much energy?

Kinetic energy is given by E=0.5mv2.  With a little math you’ll find that different perspectives of the same event result in different amounts of total energy.

Physicist: When you hear about the conservation of energy it’s natural to think of it as being something like the “conservation of chairs”: there is a total and that total never changes.  But while differently moving observers will agree on chair count, they’ll disagree on how fast those chairs (and everything else) are moving.  Velocity is subjective and therefore everything that depends on velocity is also subjective.  Including energy.

Most of the physical laws we’re taught only work in the context of “inertial reference frames” (“reference frames” for short), which is just a point of view that’s moving at a constant speed.

The chess pieces behave normally (as though they were sitting still) because they’re in an Inertial Reference Frame; traveling in a straight line at a constant speed.

Chief among those laws is the conservation of energy, which rightly says “energy can neither be created nor destroyed”.  If you get a nice rock upon which to sit and watch the universe forever, you’ll find that this law holds up: if you total up the amount of energy everywhere, that value never changes.

But the conservation of energy operates on a frame-by-frame basis, not between frames.  Someone else, drifting past at a fixed velocity, will agree that the total energy stays the same, they’ll just disagree about what that total is.  That is to say, the amount of energy in a system changes (only) when you change reference frames.  For example, if you suddenly start walking the kinetic energy of the Earth jumps tremendously (it’s a planet moving past you at walking speed).

If you’re at rest with respect to the Earth, it has no kinetic energy (never mind its rotation).  But if you’re moving, even a little, you’ll see it as having a huge amount of kinetic energy.

But clearly, very few of us are gods.  Your decision to walk across a room doesn’t induce the rest of the universe to suddenly gain and then lose energy.  The situation ultimately boils down to perspective.  When you turn your head to the right you’ll notice that, miraculously, those things that were once to your right are now in front of you whereas those things that were once in front of you are now to your left.  It’s not that the universe changes, it’s that your point of view changes.

The most that you can swing the universe around at a moment’s notice.

In this sense velocity is very much like direction or position; when you change your point of view a lot of physical things change relative to you, but that doesn’t mean that they’ve physically changed.  Asking “how do other things get new energy when I change my speed?” is a lot like asking “how do things move in front of me when I turned my head?”.  Unfortunately, the same math is used to describe both physical changes (a thing actually moves) and reference frame changes (a thing appears to move because you moved), so physicists need to take pains to keep track of which is which.

The closest you can come to an “objective measure of energy” of a system is the minimum, which you see when you’re at rest with respect to the system’s center of mass.  But even that’s pretty artificial.  It’s the answer to the question “how much energy does this thing have when it’s sitting still?”.  You could just as easily say that the only “objective distance” to any given thing is zero: the distance you measure when you’re standing next to it.

If you find yourself in the enviable position of doing physics, you generally want to pick a particular reference frame and stick with it until you’ve calculated what you’re going to calculate.  That way you can use conservation of energy and momentum at will.  It doesn’t matter which reference frame you pick, just that you stick to the same one throughout.  The energy may be different, but the physical predictions about what happens will be exactly the same.

Answer Gravy: The beauty of “advanced” physics, like Relativity, is that it allows you to recast complicated things as simple.  This is a big part of how physicists pretend to be smart.

I’m about to use linear algebra which uses buckets of matrices.  While there is a lot to learn about them, it takes <5 minutes to learn the basics behind how to use one.

First, consider momentum.  When you rotate yourself the direction of a momentum vector changes, but the length, |\vec{p}|^2 = p_x^2+p_y^2+p_z^2, stays the same.  A rotation by angle \theta in the x-y plane is described by R=\left(\begin{array}{ccc}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{array}\right).  These rotations look like R:(p_x,p_y,p_z)\to(p_x^\prime,p_y^\prime,p_z).  If this were the only kind of rotation you had access to, it would be reasonable to believe that there are two conservation laws, one for the x-y plane (since (p_x)^2+(p_y)^2 = (p_x^\prime)^2+(p_y^\prime)^2) and one for the z-direction (since p_z is unchanged).

At least until someone comes along and points out that if we include the z-direction we can write that rotation as \left(\begin{array}{ccc}\cos(\theta)&-\sin(\theta)&0\\\sin(\theta)&\cos(\theta)&0\\0&0&1\\\end{array}\right).  Suddenly x-y rotations are just a special case of a more general set of rotations (which include x-z, y-z, and any combination thereof).  Even better, the z-direction ceases to be special.  Which is good!

Something similar happens to energy when you lump the time “direction” in with the other dimensions.  In the hazy days of Newtonian physics we knew all about conservation of momentum and independently the conservation of energy.  Although, technically Newton only discovered the conservation of momentum; it took Émilie du Châtelet to figure out that kinetic energy is a thing, which is much more impressive.

Einstein, being clever, found a way to describe space and time together and in the process combined both conservation laws into one: the conservation of 4-momentum, p^\nu.  Kinetic energy is literally the time-component of the 4-momentum: p^\nu = \left(\frac{E}{c},p_x,p_y,p_z\right).  That “c” is the speed of light.  Try not to notice it.

That same x-y rotation can be done in 4-dimensional spacetime using \left(\begin{array}{cccc}1&0&0&0\\0&\cos(\theta)&-\sin(\theta)&0\\0&\sin(\theta)&\cos(\theta)&0\\0&0&0&1\\\end{array}\right).  In Relativity, moving into a new reference frame (changing your velocity) is essentially a rotation, called a “boost“, between t and a spacial direction.  It not quite the same (time is different after all), but it’s remarkably similar.

Suddenly moving in the x direction at a fraction \beta of light speed “boosts” the world using \left(\begin{array}{cccc}\gamma&-\gamma\beta&0&0\\-\gamma\beta&\gamma&0&0\\0&0&1&0\\0&0&0&1\\\end{array}\right), where \beta=\frac{v}{c} and \gamma=\frac{1}{\sqrt{1-\beta^2}}.  Just to really beat you over the head with the parallels, physicists will sometimes write this as \left(\begin{array}{cccc}\cosh(\xi)&-\sinh(\xi)&0&0\\-\sinh(\xi)&\cosh(\xi)&0&0\\0&0&1&0\\0&0&0&1\end{array}\right).

Ordinary rotations leave the magnitude of ordinary momentum, |\vec{p}|, fixed.  The amount of momentum pointing in (for example) the x-direction, p_x, can change, but |\vec{p}|^2=p_x^2+p_y^2+p_z^2 always stays the same.

Boosts leave the magnitude of 4-momentum, |p^\nu|, fixed.  The amount of 4-momentum that points in the time-direction, \frac{E}{c}, can change, but the magnitude of the 4-momentum stays the same.  Here the difference between regular geometry and spacetime geometry makes itself very apparent.  The length of the 4-momentum, p^\nu = \left(\frac{E}{c},p_x,p_y,p_z\right), is given by \left|p^\nu\right|^2 = -\left(\frac{E}{c}\right)^2 + p_x^2+p_y^2+p_z^2.  Why is that first term negative?  Because time is weird.  In fact, it is exactly that weird.

This is a really terse and totally insufficient summary of boosts and 4-momentum.  The point is: just like velocity, energy is subjective and changes in very much the same way that the direction of velocity changes when you turn your head.  Not exactly the same (because time is weird), but the difference basically boils down to some extra c’s and negative signs.

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29 Responses to Q: How can something have different amounts of energy from different points of view?

  1. I don’t get the use of the term subjective. The energy is different in different in different frames. That is objective fact.

    The fact that higher energy of the moving frame doesn’t produce more damage or heat is do to conversation of momentum. There has to be enough energy left so the momentum will be conserved. Only the center-of-mass energy (which is invariant) is available to produce damage.

  2. Neruz says:

    ‘The energy is different in different in different frames. That is objective fact.’

    Actually it’s subjective fact; the different frames are the subjects in question.

  3. Mike Mason says:

    It seem that it all kind of boils down to the fact that energy is not really a ‘thing’ or ‘stuff’. It is a number we calculate to describe how things and stuff will behave.



    Every thing is energy. It 4 component of a 4-vector. As real as can be. It is different in different reference frames. Conservation of the other three components means you can only use part of it to smash stuff, heat stuff, or create particles,

    The amount you can use is the energy in the frame where the momenta are zero. It not mystical, Basic mechanics.

  5. >Actually it’s subjective fact; the different frames are the subjects in question

    Huh? Reference frames are not general considered ‘subjects’ A reference frame does not know what the energy of something as viewed by a subject traveling at it’s velocity.


  6. The Physicist The Physicist says:

    @Traruh Synred
    You’re right, Reference Frames aren’t “smart” enough to know how much energy something has. However, everyone and everything that shares a reference frame (everything that is traveling in the same direction at the same speed) will agree on how much kinetic energy something has. It’s subjective because universal agreement is impossible and there’s no “correct” amount. In particular, that amount is definitely not the energy a thing has in its own rest frame.
    In their rest frames, bullets and asteroids (for example) are completely harmless because they’re stationary; zero kinetic energy according to Newton. They still have energy of the “E=mc2 variety”, but that typically doesn’t matter. A bullet on the shelf isn’t hurting anyone.

  7. David says:

    Trying to spin an answer out of what we know, as you guys are doing, is like trying to make a shirt when you don’t have enough material to make one. If only it was fashionable to say ‘we don’t know’, it should be, that’s what the great physicists admitted before they even got started, and that’s what drives good science.

    The truth is, we know the set of rules for this, but we don’t know why they exist. Trying to argue that there’s no reason, and they just are like that, fails in many, many places, and potentially works in a few. But it fails for another reason as well – it fails when you look at the massive bias in many people’s attitudes, towards the idea that if we don’t know about it, then it doesn’t exist. But I could show the existence of major unknowns to you in 30 or 40 ways of arguing it.

  8. Neruz says:

    Is it even possible to have a reference frame without at least one ‘observer’ in said frame? I’m can’t think of any real way to do that.

  9. The Physicist The Physicist says:

    You can do it mathematically (which is to say, you can imagine it and apply the laws we know).

  10. Of course you can have a ‘frame’ w/o observer. The falling tree makes noise in the forest even if nobody is there and the moon is there whether you look or not other than in late night beer fueled sophomore bull sessions.

  11. Steve says:

    I think everything is energy. Just changing from one form to the other. You have to see the whole picture and can not just focus on one frame. Steve http://igeus.biz

  12. >I think everything is energy.
    This borders on new-age gibberish though borders on being true depending on what you mean by ‘energy’ and ‘thing’. E=MC2.

    The point is that the equations are invariant under change in frame, but the values of “energy-momentum 4-vector change. This invariance is the ‘big picture.

  13. Neruz says:

    @Traruh Synred
    Given that the question is about Special Relativity, I’m using the Special Relativity definition of observer: A frame of reference from which a set of objects or events are being measured. “Observer” in physics does not necessarily mean ‘a person.’
    See: https://en.wikipedia.org/wiki/Observer_(special_relativity)

    Hence my previous musing about having a frame of reference without an observer, as in Special Relativity the two are usually the same thing. Though it turns out there -are- other definitions of observer depending on what area of physics you’re looking at; ex in QM observers mean ‘anything that takes a measurement.’

  14. I still don’t see what that means about ‘reference frames’ not existing w/o an observer.

  15. The Physicist The Physicist says:

    @Traruh Synred
    In physics we find that it’s important to be very careful about what we do and don’t think exists. In a nutshell, if you can’t directly measure it in a “it’s right there, doing exactly that” sort of way, then it’s not entirely safe to say that it exists. It might, but we’ve seen in the past that assuming that unmeasurable things exist makes an ass out of physicists (more on that).
    The problem with a reference frame with nothing in it, is that there are no useful physical measurements that can be made on it (more on that). You can talk about it in the abstract, but that’s arguably not quite what physics is for.

  16. Well perhaps ‘reference frame’ is just a concept. It exist enough for me. If you boost to one you’ll find the energy and momenta you measure there to be what you predict.

    Over at https:platofootnote.wordpress.com/ they would they would twist themselves into knots over this.

  17. I’ve been arguing for years that Schrodinger’s cat is silly. It is either dead or alive.

    I’ve written a story to illustrate the point.

    Schroedinger’s Cat and the Law: https://goo.gl/N4dBp2

    And the tree falling in forest makes a sound too …

  18. Neruz says:

    @Traruh Synred
    You can say the cat is dead or alive as much as you like; the reality is that the mathematics says that the cat would be both and so far all tests we’ve been able to conduct have proven the mathematics to be true.

    Some originally thought that there would be a size limit on superposition, but so far no such limit has been observed. And a deeper understanding of the ‘logic’ behind Quantum Mechanics suggests that no such limit is needed in order to be consistent with observed reality.

    Experiments have successfully demonstrated superposition in things much larger than single atoms, including a cloud of gas and a small strip of metal large enough to be seen without a microscope.

    So while it could arguably remain possible that there is some size limit, there are no theoretical nor experimental reasons to believe that is the case. The opposite is true; all available evidence emphatically supports unrestricted superposition.

    Yes, that flies in the face of common sense and normal logic. Common sense is wrong. Which isn’t really very surprising as human brains definitely did not evolve to understand the fundamental underpinnings of reality. They evolved to find food and make tools out of wood and stone.

    You are correct in one aspect though; Schrodinger’s Cat -is- silly. It works only as a thought experiment and even then not very well, in reality there is no known practical way to put a cat into a state of superposition like that. Schrodinger’s Cat being silly does not in any way make Quantum Mechanics any less true however.

    After all, the universe is (as far as we know) a thing that just sort of happened. No-one planned it or designed it, there was no logic or reason behind its structure. Why should something like that make intuitive sense? I would expect the details of such a structure to be confusing and chaotic, neatness doesn’t happen on its own.

  19. Neruz:

    A Cat is not in a quantum state. It is constantly interacting with the environment – being measured!

    It is either dead, alive or perhaps dying.

    Putting it in a box does not make any difference — other than making it easier to figure out when it died. E.g., Just measure how much oxygen is left.

    Schrodinger’s cat is not to be taken seriously and was not intended to be.

    Read the story.

  20. Nomad says:

    Hi, (a newbie here, though not totally newbie to the science field and note that english isn’t my best language, will try).

    E=a constant (including m, scalar) x v^2 (square of a vector, velocity) = E = a scalar quantity OR a vectorial quantity? (of course, it E is a scalar quantity, but, why did I write so with a question mark ? It is because…)

    it is said here that velocity is subjective/relativistic qty, so is E.. This reasoning isn’t good enough, because, yes, velocity v is a vectorial qty, i.e. has a direction which is related to referance frame, hence, subjective, relativistic, etc, but, the velocity term in energy equation is “square” of velocity, that is, a scalar qty. That’s, its character of velocity is changing, like it happens in i^2=-1 in which “i” is implicitly related to direction (circular) while its square is a constant line.. So, actually, E is not a subjective/relativistic quantity which changes wrt a reference frame, i.e. conserved in any frame.

  21. Neruz says:

    @Traruh Synred

    But if you put it inside a theoretical Schrodinger’s box that was somehow able to prevent it from interacting with the environment, it would most likely enter a state of superposition. Hence why the cat only works in theory and is both silly an correct; we have no way to make such a box.
    (Technically from the right perspective it, along with everything else, is in fact in a state of superposition all the time. It’s just that we’re entangled with the cat and thus we see only one state. There are also other issues with the thought experiment, as it treats the observer and the measurement process classically. Which cannot possibly be correct, Physicists and their apparatus must be bound by the same quantum mechanical rules as everything else in the universe.)

    I am aware of the story behind Schrodinger’s cat; Schrodinger intended the thought experiment as a way to show how the Copenhagan interpretation of Quantum Mechanics was far too absurd to possibly be correct. (Hence the flaw regarding the nature of the box.) The irony of the story is that despite this, the thought experiment turned out to be generally correct in a theoretical sense.
    It also has the added bonus of being easier to understand than the more accurate modern interpretations of Quantum Mechanics, which is why it’s taught in schools alongside similar ‘close enough for most people’ physics lies like atoms being clusters of balls in discrete locations.

    Despite Schrodinger’s (and Einstein’s) annoyance at the inexplicably confusing nature of Quantum Mechanics, the Copenhagen Equations -work-, definitively. And experiments have put items with as many as 10 trillion atoms into a state of superposition.
    The extra double irony of the story is that while the Copenhagen interpretation -was- flawed, Schrodinger and his peers focused so intently on the seeming absurdity of the situation that they completely missed the real flaws, such as the aforementioned ‘treating the observer as separate from the system being measured’ issue.

    Does this mean a cat can also be put into a state of superposition that we can somehow measure? No, but it does mean that so far no upper bound on size has so far been found and until that upper limit or evidence of such is found we must necessarily assume that there isn’t one. If there isn’t an upper limit, then anything can theoretically enter a state of superposition and indeed everything must necessarily be doing so pretty much all the time.
    We just can’t notice it, because we’re inside the wave function and thus can only see one possible outcome of its collapse.

  22. >But if you put it inside a theoretical Schrodinger’s box that was somehow able to prevent it from interacting with the environment, it would most likely enter a state of superposition. Hence why the cat only works in theory and is both silly an correct; we have no way to make such a box.

    In such a box the cat would be quickly dead; it would not live long enough to enter a dead/live superposition. Life itself is a matter of interacting with the environment. So is death. Life and death are not defined on a femto-second time scale or even nano-second.

    It’s all silly, even though folks like John Ellis seem to take it seriously.

    You can have a version in many worlds, but which ever world you end up in, the cat is dead or alive. The dead and live worlds are orthogonal and nevermore affect each other after the irreversible that splits ’em.

  23. Neruz says:

    How long the theoretical cat can survive inside the theoretical magic box depends on what else is in the box as well, and how big it is. If the cat is dying too quickly it just means you need a bigger box, and since this is an imaginary thought experiment with magic boxes the box can be arbitrarily large with no problem.

  24. A big box just makes it more difficult to determine when the cat died.

    The cat still interacts with it’s environment whether in a box or not. A cat that does not interact with it’s environment [a] is already dead. Even a dead cat interacts with it’s environment if only by cooling (thus allowing time of death to be estimated).

    [a]I thought the ‘magic’ box prevent any interaction, thus allowing a QM state of the cat to be defined.You now seem to be talking about a ordinary box.

    Boxes are red-herrings.

  25. Neruz says:

    It doesn’t matter if the cat interacts with the environment inside the box; the box is a magic box that somehow prevents\removes any kind of entanglement between its contents (the cat+enough air to keep the cat alive for the hour or so needed to conduct the experiment) and the things outside the box (the scientists and measuring tools.)

    The cat is of course entangled with everything inside the magic box, but that is irrelevant because from outside the box the only thing you can tell is that the contents of the box are in a superposition of states.

    Although AFAIK there have been a few experiments conducted which managed to measure a qubit without collapsing its superposition, so there’s probably some theoretical way to find out what state the cat and other contents of the box are in. I am not sure how though, but given that this is a thought experiment that already involves a magical box that’s not really a problem.

    You seem to be overly focusing too much on the ‘literal reality’ of the cat and missing the underlying logic of the thought experiment; thought experiments are not meant to be taken literally like that. That’s why they are thought experiments; if a thought experiment can be taken literally then it’s not a thought experiment, it’s just an experiment.

  26. >It doesn’t matter if the cat interacts with the environment inside the box; the box is a magic box that somehow prevents\removes any kind of entanglement between its contents (the cat+enough air to keep the cat alive for the hour or so needed to conduct the experiment) and the things outside the box (the scientists and measuring tools.)

    Huh? This makes no-sense. Schrodinger did not specified a magic box — merely a sealed’ one. Sealing the box is what makes it so easy to figure out when the cat died. Even if supplied with enough air for ‘the hour or so’ the cat will breath it and the ratio of o2 to co2 will change. Not being entangled with the outside is irrelevant. Schrodinger’s cat is not about entanglement.

    A cat is interacting with it’s environment. It changes that environment. Even if prevented from detecting what’s occurring in the box, traces will be left.

    A cat both dead an alive would indeed be magic.

  27. The Physicist The Physicist says:

    @Neruz and Traruh Synred
    I really appreciate this tete a tete.
    You may be approaching this from different first principles. It may boil down to answering “when does the collapse kick in?” and “what does the box do?”. Correct me if I’m wrong, but it sounds like Traruh Synred is thinking that when the cat (read “quantum system”) interacts with an environment of any kind, it assumes a particular state; either alive or dead. Neruz seems to think that the collapse of the cat’s state waits until the magic box is opened.
    This definitely isn’t an experiment that can actually be done (hence the magic boxes). The box is a way of saying “these systems definitely don’t interact in any way”. We can consider the cat, the interior of the box, and the world outside of the box as three quantum systems. The cat and the interior of the box can interact and (assuming we’re following the rules for actual, real-life quantum systems) the combination of the two can be in multiple states. Simply interacting with another system isn’t enough to “collapse” the cat into a single state. Instead, when pairs of otherwise isolated quantum systems are allowed to interact with each other, they become entangled (read “still multiple, but now correlated states”).
    From an outside perspective, the cat/interior-of-box is in an entangled state: a superposition of “living/clean” and “dead/smelly”. Opening the box (interacting with the quantum system) entangles you with the cat/interior, but the story is more or less the same. If you and the magic box were in an even bigger magic box, then (presumably) someone else would see the cat, the interior, and the scientist as a single cat/interior/scientist system that is also in an entangled state: a superposition of “dead/stinky/horrified” and “living/clean/relieved”.

  28. >a superposition of “living/clean” and “dead/smelly”.

    Well it would have to be a superposition of more than dead/alive, but alive til now(with appropriate amounts of CO2 added and O2 removed for each live state, and dead for how long (how ‘smelly’), bacterial load, etc.

    This leads toward many worlds in which after each ‘irreversible’ event the world splits into states that no longer can interfere with each other. They are still ‘superimposed’, but in your ‘world’ the cat is dead or alive.

    Now if you want collapse you can move the collapse between the first and irreversible event and ‘observation’ w/o changing the results. If you make it ‘observation’ then you have a kind-of temporary many worlds, that collapse to one when you look. Seems kind of silly to me, but not inconsistent with available data (and the pathologist can still tell when the cat died).

    To me it seems simplest but the collapse at the earliest irreversible event, which likely happens faster the neural signal can transmit up the cat’s tail.
    A cat living or dead is not in a quantum state, but transiting among many of them. In collapse picture near continuous collapses are part and parcel of being alive (respiratio or indeed dead (decay-and-smell). ‘Smelly’ is not a QM state. Irreversible interactions release nasty gases that since the box is sealed accumulate and could be measured.

    This is the basic point of ‘Schrodinger’s Cat and the Law’

  29. The Physicist The Physicist says:

    @Traruh Synred
    Schrodinger’s Cat is a simplified two-state case; like you say there’s a lot more than 1 bit of information. But keep in mind, the past can also be in a superposition of states (see “bell inequality for position and time” by Franson). Measuring what state a thing is presently in follows the same rules as determining what state a thing was in.
    The cat can certainly drop dead at any moment and (in this idealized experiment) until the box is opened its not just in a super position of living and dead, but a superposition of living and having died at every time.

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