Special Relativity: Where and When

In Aristotle's space-time there as an absolute notion of positions and motions. In Galileo's space-time we lost absolute positions and motions, but we retained absolute times. Saying when two events occurred with respect to one another was objective and absolute. In my last post I had two example diagrams featuring two astronauts flying past each other in empty space. The diagrams were equivalent in the sense that they depicted the same state of affairs, only differing by a coordinate transformation. There was no objective answer to the question whether you were moving past your stationary friend, or vice versa. But there was an objective answer to the question of when (for example) you two crossed paths, or indeed, the order of any sequence of events either of you witnessed.

This latter part---questions about when things happen, and their order, will lose their objective status in Einstein's relativity. And we'll see why this causes trouble for faster-than-light travel in science-fiction stories.

To begin we need to come up with a concrete understanding of what "happens at the same time" means, or even better, how to test it. For this we need a protocol for determining when events too distant for us to see "happen" relative to our own lives. To give a feel for this I will shamelessly self-plagiarize from my dissertation:


Imagine sending an RSVP to a friend, Ali, and you'd like to determine when, exactly, they get the invitation. We first make three simplifying stipulations: (1) It takes no time for your friend to compose their a response, (2) they will do so immediately upon receiving the invitation, and (3) the invitation spends no time sitting in the mailbox (i.e. the mail carrier immediately picks it up at both ends). Now, you send the invitation at noon on Monday, and receive Ali's response at noon on Wednesday. From this information we can easily calculate the total elapsed time: 48 hours. From symmetry assumptions (specifically, that there is no difference in the average speed of the mail carriers nor is there a difference in distance traveled on the outbound and return routes) we can then calculate that the letter arrived at Ali's house at noon Tuesday (that is, exactly 1/2 the round-trip time). You can now say with confidence that Ali received notification of your party exactly when you were enjoying your lunch on Tuesday. Now, suppose that you sent out invitations (at noon) in the same manner on the previous Saturday and Sunday to Boris and Claire respectively. And further suppose that you received a response from Claire on Thursday at noon, and from Boris on Friday at noon. By the same methods described above, you would calculate that they too both received their letters at noon Tuesday (a remarkably coordinated mail service!). Thus, the events of Ali, Boris, and Claire receiving invitations, and your eating of lunch on Tuesday all occupy the same "simultaneity space"---we say they are simultaneous with one another.


Let's summarize the protocol---to figure out when a distant event happened simultaneous with some event in your life, first, send out a signal that can be returned by the event in question. Second, start tracking to time, and record the elapsed time it takes to receive the returned signal. Finally, divide that total time by two---whatever you were doing half-way between sending and receiving the signal was simultaneous with it.

Luckily, pictures help, and again I will  copy from my dissertation:

first_diagram.png

What we see here is a space-time diagram of an object alpha. At event s alpha sends a flash of light off---shown above by the dotted line. After that event alpha waits, eventually receiving a flash of light from event e at r. Looking back on the time passed between the sending and receiving of the signal, alpha reasonably divides the total time in half, and concludes that their signal was received by e at the same time that alpha was experiencing event o. Of course, there was no way for alpha to know that events o and e were simultaneous at the time, but that's no impediment to concluding post-facto when the two events occured.

That's all well and fine, and it comports perfectly well with our common sense notion of 'simultaneous'. Things change, however, then we compare two observers in motion relative to one another:

SpaceTime_2_Alpha.png

In the diagram above, we see an observer beta in motion relative to our inertial frame---specifically moving from left-to-right. Notice that beta is performing the same measurement protocol for determining simultaneity that we described for alpha. However, the principles of relativity begin to have a noticeable effect. Recall that one of the fundamental assumptions which undergirds special relativity is that the speed of light is constant under all inertial reference frames. What this means for the diagram above is that even though beta is moving from left-to-right from our vantage point, when she sends out a flash of light at event s, that light signal doesn't receive a 'boost' from her velocity, like we might naively expect. This is represented in the diagram by noting that the light beam remains at a 45-degree angle relative to the diagram's overall orientation.

As beta carries out the measurement protocol we see that her 'simultaneity-space' (the collection of events mutually simultaneous) is 'tilted' relative to our perspective. She will determine that e happened simultaneous with an event noticeably prior to event o. It's due to this sort of behavior of measurement by light signals which results in our claim that the time-ordering of events loses objective meaning in special relativity.

We can see this explicitly by considering two observers in relative motion in one space-time diagram:

SpaceTime_3.png

Here we have both alpha and beta, with alpha stationary in our reference frame, and beta move to the right. Suppose they are collocated at event s and both agree to send off light signals. For alpha, the protocol determines that events e and e' are simultaneous, as will be every event falling in the simultaneity space Aα. However, for beta the time ordering of events will be quite different. Event e will happen a significant time earlier than e'. Additionally, it's not hard to see that a third observer moving to the left, and also collocated at s will see e' happening before e!

Next time we'll see how this feature (namely, relative motion changes your simultaneity space) will cause trouble for FTL fleets looking to invade distant solar systems.

Galilean Relativity: A primer on space-time

Last time we saw how to picture Aristotle's world with space-time diagrams. To review: the positions of objects at any particular time can always be specified relative to the origin and coordinate system of the background space. If Hilde is 10m from the center of the universe, then that's a fact, and whether anyone knows it or not, it's true.

We also saw that we can depict changes in position over an interval of time (i.e. motion) by using space-time diagrams. A stationary object is represented by a vertical line (no change in position per unit time), and angled lines can tell us how fast an object is moving. 

In Aristotle's universe there is only one correct way to view such a diagram. There we have a notion of absolute rest, and absolute motion. In the Galilean space-time we'll look at below, we trade absolute motion and position for relative motion and position.

What does this mean? Galileo's idea was that there is no observable difference between being motionless, and moving at a constant velocity. If you were in the hold of a ship on perfectly calm seas, you would have no way of knowing whether you were still tied to the pier, or sailing along. A more modern example might be the experience of sitting on an airliner. In calm, straight flight, there is nothing (other than the sound of the engines perhaps) to tell you that you are moving save for the view out the window.

Motion, in Galilean space-time, is relative---I can only specify my velocity with respect to something else. There is no such thing as absolute velocity (or position for that matter). Intuitively this might seem like bad news---it seems like we lose the ability to state as many facts about the world. That is, according to Aristotle, there's a well-defined answer to any question about where anything is in the universe, and how it's moving. We lose that ability in the new framework.

What facts remain? It turns out that while we lose the ability to say anything about absolute velocity, the relative velocities we measure are invariant. What does this mean? Suppose that you are floating in empty space, with no objects or landmarks that you can see. There is no way for you to determine whether you are moving or not, and indeed, Galileo tells us that there is no absolute fact of the matter. Now suppose from the distance you see another astronaut hurtling past, from left to right in your field of view. Are they moving past you, or are you moving past them? Again, there's no absolute fact of the matter in this case, but that doesn't mean we can't say anything about the motions involved.

We can represent the two astronauts (you and your friend) in a space-time diagram. In Figure 1 let's suppose you are the pentagon, and your friend is the triangle. In this diagram you are represented as being at rest, and your friend is moving with a constant velocity (say, 10 meters per second). 

 Figure 1: A picture of you (the pentagon) at rest.

Figure 1: A picture of you (the pentagon) at rest.

However, if we imagine the situation from the point of view of your friend we can see that their situation is largely the same. They experience someone (you) zooming past them. This can be represented by a transformation of the space-time diagram. I've drawn this in figure 2---notice that here it is the triangle which is depicted at rest, with the pentagon traveling past it.

 Figure 2: The same situation from a different perspective.

Figure 2: The same situation from a different perspective.

These two pictures are very different. However, there's an important feature which does not change: the relative distance between the two objects at any horizontal 'slice' of time. A consequence of this is that it is easy for anyone to calculate what each person will measure. In this particular case, if you measure the other astronaut passing by at 10 meters per second, they will measure you passing them at 10 m/s (in the other direction). Motion, in this framework, is only relative, but those relative motions will be objective for any observer.

To sum up, the move to Galilean space-time requires giving up the ability to say certain things definitively---there's no longer any fact-of-the-matter about an object's absolute velocity. But we find there remains some shared facts which remain invariant for everyone who is measuring the system (like the difference between any two object's velocities). Next, as we turn to special relativity we'll find a similar pattern---certain features which seem to be objective (or 'absolute') in Galilean space-time will be relativized in Einstein's theory.

 

Aristotle's Space-Time: A primer on space-time theories (pt 1)

Before we look at special relativity, it will be helpful to look at the historical theories of space and time which it replaced. With these in the background it will become easier to see what assumptions are being made in any of these theories, and how the picture of space and time change with each new theory.

In order to start, we need to learn how to read a "space-time diagram". This is really just a fancy word for a graph, specifically which plots a spatial dimension against a temporal dimension. We're all familiar with this concept, even if it's been a few years since our high school physics class. In the graph below (fig 1) time is plotted on the x-axis, and distance (or position) is plotted on the y-axis. The little triangle points represent the movement of a particle as a function of time.

 Figure 1: A distance-time plot as you might see in your physics textbook.

Figure 1: A distance-time plot as you might see in your physics textbook.

In general, the way to read graphs like the one above is to imagine that each step along the x-axis (time) is like a snapshot of the state of the system in question at that moment. We represent a dynamic system (i.e. one that changes) by imagining a series of static systems progressing in time. This is precisely how we'll represent space and time in the coming discussions. However, physicists are creative folks, and not content to plot space and time as we typically do, with time on the x-axis, they flip the axes, so we think of time as going "up" and space as going "left" and "right" (fig 2).

 Figure 2: The same plot as above, but with the x and y-axes exchanged. The object is moving to the left and speeding up (since it covers more horizontal distance per vertical tick). Labels also flipped for dramatic effect.

Figure 2: The same plot as above, but with the x and y-axes exchanged. The object is moving to the left and speeding up (since it covers more horizontal distance per vertical tick). Labels also flipped for dramatic effect.

This plot will be our most fundamental tool for examining theories of space and time. I have a bit more to say about such diagrams, and how to think of them, but it will be useful to think about them in the context of an actual theory.

Aristotelian Space-Time

For obvious reasons we will consider theories of space-time which are limited to two dimensions: One spatial, and one temporal. The universe in which we will be examining these theories consists, in its spatial extent, in a straight line. That's it. Organisms in this universe have two choices when they want to move: left or right. If it helps, you might conceptualize such creatures as tiny ants on a clothesline, pacing back and forth, but with no other options.*

 Figure 3: The universe.

Figure 3: The universe.

Let's meet two of these creatures, Anselm and Baruch. They both live somewhere in the universe, and we can represent that by drawing a point on the line.

 Figure 4: The universe, now with Anselm and Baruch.

Figure 4: The universe, now with Anselm and Baruch.

How do we know where to draw these points? How far apart are they? How might we measure in order to know if we've got the drawing right? It is exactly these questions it turns out, which motivate a theory of space-time! That is, how we answer these questions, what assumptions we make, will differentiate Aristotle's space-time from Galileo's, and Galileo's from Einstein's.

For Aristotelian space-time we pick an origin and a standard length measure. With these two things we can impose a coordinate system on our space (fig 5). The important thing to notice about our treatment of space is that the origin is fixed. There is, on this picture, a correct answer to the question 'who is closer to the center of the universe?' If the origin is located as it is in figure 5, then the answer is Anselm. **

Things get more interesting when we start talking about how things change in time. Figure 5 only shows the positions of Anselm and Baruch at a given instant. By looking at this picture there is no way to tell if they are moving, and if they are, in which direction and at what speed?

 Figure 5: The universe with an origin and a coordinate system.

Figure 5: The universe with an origin and a coordinate system.

 Figure 6: 1 second later

Figure 6: 1 second later

One typical way that we understand the motion of objects is by seeing their position change in time. But this requires animation, and isn't suited for drawings (and blog posts). Instead, we will return to our space-time plots from above.

Consider figure 6, which is much like figure 5 but Anselm's position is different. Suppose I told you that this figure was a snapshot taken 1 second after figure 5. We now have much more information about what's going on. We know that Baruch is (virtually) stationary, and that Anselm is moving toward Baruch (to the left) at roughly one unit per second.

 Figure 7: Consecutive snapshots of our universe.

Figure 7: Consecutive snapshots of our universe.

If we take more of these snapshots, each 1 second apart (fig 7), we can begin to see a clearer picture of how Anselm is moving. We can imagine, rather than taking snapshots, we just let the position of our objects continuously record as we scroll upwards (fig 8). This upwards scrolling is what allows us to represent time in a drawing! And here we finally have it, a space-time diagram:

 Figure 8: B stands still while A moves to the left.

Figure 8: B stands still while A moves to the left.

A few comments are in order before moving on. First, the objects A and B (i.e. our friends Anselm and Baruch) should be thought of as represented by the entirety of the (magenta and green, respectively) lines. We will call these 'world-lines', and they mark out the series of events which together constitute the object. Second, events are the points on the worldline---they are the instantaneous 'happenings' which strung together make up a persistent object. While this is perhaps a rather odd and foreign way to think about objects, you will find you quickly adapt.

This picture of space-time has some intuitive features which will no longer hold in the coming theories of space-time. To end we'll look at one in particular. Notice that in figure 8 there is no way to read this diagram as saying Anselm is at rest while Baruch is moving.*** There is an objective fact of the matter about who is moving, and more precisely, at what speed, and in which direction. This seems entirely reasonable! When I stand on the sidewalk and see cars drive past, it is as clear as day that they are the ones moving, not me.

What is surprising is that our ability to make such distinctions in Aristotelian space-time relies wholly on the fact that we posited that there was a single, fixed, objective origin. On this picture of space-time it is always possible to determine the single, unique, position of any event, as well as the speed and direction of any object. This is made possible by the fact that everyone must agree on the origin.

As we'll see next time, even though this intuitively seems plausible, there are equally plausible considerations against there being such an origin. We will then begin to see what, if anything, we can say about objects and events in such a space-time devoid of a center. This will be Galilean space-time.


*You might wonder, "What happens when the ant gets to the end of the line (that is, goes all the way left or right)?" Great question! We're going to ignore it for this series of posts, but there's no reason why it wouldn't make a great topic itself.

**The assumption that is being made in this case is that the space (i.e. the one dimensional line) has some sort of real, physical existence. The idea is that we could, in principle, determine every object's true position relative to the origin if only we knew where it was. This commitment is at the heart of the Aristotelian picture.

***The distance between them is shrinking so at least one of them must be moving!

Special Relativity for Science-Fiction Authors

Albert Einstein is widely known for being the genius who first articulated our modern physical theories of space and time. For most people this has secured his status as an unparalleled mind, achieving more and seeing farther than most could hope to even imagine. There is one segment of society, however, for whom Einstein's notoriety comes not from his brilliant and uncanny mathematical and physical intuition, but for the havoc and despair which the fruits of that intuition have delivered.

I'm speaking, of course, of science fiction authors. The special and general theories of relativity have been, for as long as they've been articulated and understood, a thorn in the side of exciting, dashing, romantic, fantastical space adventures. Even those who know nothing about the details of special relativity (SR)* know the basics of the problem it poses: "Nothing can go faster than the speed of light." you might hear someone say. And while many are familiar with (some of) the limits SR imposes, I would guess that many fewer have a good grasp of (1) what the problems are which those limits impose, and (2) why the limits cause these problems.

In the absence of such an understanding, writers and readers are sometimes disposed to simply wave away the problems, or to gerrymander some special tool, substance, or circumstance which permits violation of the limits SR requires. And doing so is perfectly OK! "Science in service of story." is as good a motto as any. Plenty of lovely fiction rides roughshod over SR, either ignoring it or ignoring certain, fundamental, aspects of the theory. However, I think with a little time and effort invested, one can gain an understanding of the limitations which SR imposes, how they arise, and what they imply. Given such an understanding I strongly believe that one can better navigate their way around such limitations, in a way that is plausible and (mostly) consistent with the physical laws as we know them. Indeed, I think that understanding SR can give rise to interesting story and plot itself!

I plan, over the next several weeks, to begin to lay out a basic picture of special relativity. These posts will aim, by their end, to (1) offer up a clear explanation of what SR "is" and how it imposes limitations to how we travel through space; (2) show the variety of ways in which choices of how FTL works in science fiction can run afoul of SR; and finally (3) examine how we might "write around" the theory, doing as little violence as possible to SR, while still retaining a framework which would make for interesting stories.

The posts are planned as follows:

Aristotle, Galileo, and Einstein, a primer on space-time theories

[UPDATE: I lied! This will be broken up into two posts, one on Aristotelian space-time, and one on Galilean space-time.]

  • We'll begin by reviewing the basics of how Aristotle, and Galileo modeled space and time. We'll also see that all of our understanding can be represented in drawings.

Special Relativity via geometry

  • Using the drawing conventions we learned last time, we can set out the Special Theory of Relativity.

Consequences/Paradoxes of SR

  • We will use our space-time diagrams to explore some of the consequences of SR including some rather odd results.

FTL and SR

  • Finally, via space-time diagrams we can consider "Faster-than-light" travel and explore the variety of difficulties which arise.

"Writing Around" SR

  • To finish we think about different ways we can avoid or mitigate the difficulties, while exploring how and whether interesting stories can fit within these approaches.

My hope is that these will be edifying and entertaining. I really do believe that understanding these limitations can help in thinking of interesting, story-conducive ways of getting around them. Too often SR is held up as some sort of arbitrary stricture, with no real explanation or understanding as to why it limits what it does. Seeing it as arbitrary encourages authors to think they can just as arbitrarily ignore it. I hope these posts will help to dispel this idea.

That all said: IGNORE IT IF YOU LIKE! I love Star Wars and Star Trek. Good stories don't need to slavishly adhere to physics. I'm not here to police science fictional FTL---only to offer some tools to those who wish to do FTL slightly more realistically.

 

* As the title of this post suggests, I will be focusing the discussion exclusively on special relativity. This is for two reasons: First, the basic limitations that relativity imposes on space travel only require an understanding of the special theory (which is both conceptually and diagrammatically simpler). And second, I would be lying if I pretended to have anything like an authoritative grasp on GR---my understanding, as sketchy as it is, is best qualified to discuss SR. Perhaps a discussion about what GR adds, and my corresponding ignorance can be had in some future post...

Is philosophy useless?

I just recently ran across the following quote, from a physicist I respect greatly (and no, it's not Richard Feynman): "Maybe physicists would complain less that philosophy is useless if it wasn’t useless." The original post is here (this quote is the very last sentence of the post).

Dr. Hossenfelder is hardly the first physicist to say such a thing, and I think it's pretty clear that the above quote is for rhetorical effect since one of the very first things she says in the post is that "...I think it’s an unfortunate situation because physicists ... could [use] help from philosophers."

Right away then, it seems like two things are being conflated in that first quote. On the one hand, Dr. Hossenfelder is claiming that what current philosophers produce is useless, but on the other hand she seems to accept that philosophy as a field has value and could potentially contribute to questions faced by scientists (namely, theoretical physicists).

[She does, later in the post, say some things about how philosophy only applies to 'pre-science' and is useful only when it develops such an area to the point where it becomes useless. I think that's not quite right, but I'll leave it for another day.]

So, if philosophy is---in principle---useful, but as now practiced useless, the question that naturally arises is 'how do we get from here to there?' That is, how should philosophers (of science mainly---I'm presuming Hossenfelder has no truck with ethicists and logicians) do philosophy so that it isn't useless?

I'm a philosopher, not a scientist, so I have no special authority to answer this question. I would welcome suggestions from practicing scientists! What I will do (in typically philosophical fashion) is respond with my own question: "Useful for what?" And here is where I'm going to do that thing physicists supposedly hate---split semantic hairs.

To say that philosophy is useless without telling me what it's useless for is unhelpful. Philosophy is useless for mopping floors or making an omelette. It might also be useless for devising, building, and calibrating measurement devices (I don't actually believe this, but I'm happy to grant this for the time being). 

Here are some things I think philosophy is useful for:

  1. Ouroboros-like, examining what inquiry (scientific and philosophical) is useful for at all.
  2. More specifically, articulating for ourselves and others why the project of science is valuable.
  3. Providing a place to actively question assumptions and received truths which are not actively questioned in scientific or everyday contexts.
  4. Helping to remind ourselves of how very little we know.

These four are not exhaustive of philosophy's utility, but I have a personal interest in each:

As for (1), it is an open question why we should do anything beyond living a hand-to-mouth subsistence lifestyle with the aim of reproducing. Now, I expect anyone who reads this has an answer to that question: fine. I do too. The point is, we don't all agree on what the answer is, and even to the extent what we think others are wrong, we'll probably disagree about why they're wrong as well. This conversation belongs to philosophy. You can refuse to enter into the conversation, but you can't say that the conversation is pointless or meaningless without unwittingly taking part.

As for (2), I think this question is one of the most central, deepest, and largely ignored questions of the modern world. We're exploiting our remarkable theoretical and empirical successes to do all sorts of things (build smartphones, write blogs, construct skyscrapers, cure illnesses, etc.). Is this what makes science valuable? Is it the only thing that makes science valuable? Is our *understanding* of the world important too? Does the knowledge that everything is made of innumerable tiny things which combine in complex and surprising ways make us better, or let us have better lives? And what does it mean to have a better life? Again, no doubt, many will have an answer to this question---but let's not pretend that everyone agrees, or that all but one answer is completely indefensible. 

Moving on to (3), I'm reminded of a couple of experiences of my own from several years ago. I was taking a (philosophy) class about relativity (taught by the wonderful John Manchak). There were a number of physics majors in the class. At one point the question of interpreting a part of the theory came up (unfortunately I can't remember any of the details, all I can recall is that it was---I think---about positing an unobservable absolute space) and several of the physics students announced the choice was clear---by Occam's razor the simpler theory is the correct one. One of the philosophy grads in the class pointed out that Occam's razor isn't an inviolable truth, and it would require justification! The physics students were nonplussed to say the least. The point here is that while the physicist is well served in the practice of physics to accept Occam's razor, it can and should be open to examination and the question of justification! And as it turns out, evidence for the razor is perhaps not as robust as one might presume (don't get me started on the problem of induction!).

There was a second incident in the class that sticks in my mind too. We were busy working away, and dealing with some question or other (again, regrettably I forget the specifics). In examining whatever it was, a grad student (in philosophy, but trained in physics) pointed out that some answer would violate the conservation of energy, and so we could discard it. It was pointed out that the conservation of energy was not some a priori logical truth*, and there was no reason (for our purposes) to presume it was true. At this point I witnessed the closest thing to a brain short-circuiting I've ever come across. The student couldn't comprehend this possibility! Again, if scientists constantly questioned the conservation of energy then we'd never get anything done. But it does not follow from this fact that it's useless to think about the possibility that such a principle is in fact false (also, I'm not claiming no scientist ever does question such conservation principles!). This idea, that thinking about the possible falsity of our most empirically well-confirmed theories is something worth doing, leads to the fourth thing philosophy is useful for.

(4) The success of science (by which I mean, specifically, its empirical predictions) and the proliferation of its applications work to instill in us the sense of its epistemic impeccability. But the practice and history of science also pulls in the other direction (this is what Kuhn called 'the essential tension')---overturning our closest and most foundational beliefs, revealing the intelligibility-defying character of parts of the world closed off to our senses. We have a choice about whether to take science's defining feature as the body of knowledge it delivers, or as the attitude and technique which reveals the depths of how little we know and understand. Science does both. But I think it's healthy to take the latter view seriously, that one principal value of science is that it shows us that, and how, we should give up our beliefs (I'm shamelessly stealing this particular phrasing from Bas van Fraassen's Empirical Stance, but I don't have the book with me, so I can't cite the page---sorry!). Science doesn't and can't do this alone, and this isn't the sole utility of science by a long shot---I very much like my smartphone and penicillin. But thinking philosophically, asking about reasons, assumptions, possibilities; imagining alternatives and questioning received truths, these things are (I claim) useful. It's just that they're not useful for delivering knowledge---I am of the opinion that philosophy is simply not an epistemic enterprise. Instead, they're useful for helping disabuse ourselves of the seductive idea that we know it all.

 

Ok, well this was a longer post than I was planning. To sum things up, philosophy is useless for some things, and many of those things are things which scientists need. But philosophy is useful for other things, and these too (I think) are needed by science, even if practitioners don't agree.

Will philosophy answer whether reality is composed of strings or a space-time lattice? No. Will it be the medium through which we can coherently ask the questions of whether our concepts are adequate for understanding such claims, or whether we need new concepts, or whether such claims are so far removed as to be meaningless or ... ? The answer to this, I think, is yes. Of course this is in part motivated by a desire for job security. But it also motivates why I want a job doing philosophy at all---I think that science is the best bet in the house, and I want to understand why.

 

*And yes I'm aware of Noether's theorem---it establishes it that a time translation symmetry guarantees (explains?) the conservation of energy. However, the claim that there is a time translation symmetry is on an epistemic par with the conservation of energy, and so while this theorem provides a deeper understanding of conservation laws, it doesn't prove that energy is in fact conserved.