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Wanting to understand the one-way speed of light problem in more detail please

  1. Jul 30, 2011 #1
    I wanted to understand the issue of measuring the one-way speed of light in a bit more depth. I have seen this subject pop up in a few recent threads, however as my questions were mostly off-topic, I thought I should start a new thread rather than continue to hijack others.

    I also need to say that I in no way want to debunk SR, I am just trying to understand some of the fundamental principles in more detail. I should also say that I don’t have a background in physics and am still learning the math part, so please forgive if my approach is sometimes elementary!

    As I understand it, we do not currently have a way to measure the one-way speed of light that is in keeping with the principles of SR. So if we could measure the one-way speed of light, it would dis-prove SR. So the one-way speed of light is currently assumed to be the same in all directions, but we can’t validate that for certain.

    So I wanted to understand if the issue is:

    a) SR does not allow for the one-way speed of light to be measured (Due to the clock synchronisation issue) and that's that.


    b) We just don’t have a way of testing if light travels at different speeds in different directions.

    I saw points a) and b) as being subtly different. That is that the first point is about measuring a speed and the second point is about measuring a difference.

    My thinking is that if it was about just about proving that there is not a difference, then maybe that helps eliminate the clock synchronisation issue.

    So I was actually thinking of a thought experiment that might be able to show there is no difference, but I wanted to understand the issue in a bit more depth before I make a fool of myself again! :smile:
  2. jcsd
  3. Jul 30, 2011 #2


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    I would say that it is a).

    The point is that synchronization is a convention. We can think of synchronizing clocks using Einstein's convention, but we can also think of other conventions. For example we could think of synchronizing our clocks by using some master clock which broadcasts the current time and the distant clocks simply set their time to coincide with the master clock. In that case the "outward" speed of light would be infinite and the "inward" speed of light would be 1/2 c. We could re-write Maxwell's equations in terms of this synchronization convention and get a theory which agrees with all of the experimental results. In particular, the round-trip (aka two way) speed of light would be isotropic and equal to c.
  4. Jul 30, 2011 #3


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    If we just consider our situation without Einstein's Theory of Special Relativity, that is, without his clock synchronization convention, then it is impossible to know or observe or measure the one-way speed of light. That is because once light leaves us, we can't know where it is in it's trajectory across space. We don't know when it arrives at any particular location.

    Prior to Einstein, there was the common belief that light was an undulation (or disturbance) of the fixed ether medium. If this had proved to be the case, then light would travel at a fixed speed always relelative to this medium and not relative to an observer. This is the situation with waves traveling through other media like air, water, or solids. It should have been possible to detect the stationary state of this fixed ether medium in the case of light but all attempts failed. So the scientists of the time "explained" the null results of their experiments by postulating that the lengths of their physical apparatuses contracted along the direction of the earth's motion through this medium (although they didn't know what the actual contraction was because they couldn't detect the fixed state of the medium) and by postulating that all time related devices slowed down at the same time.

    This culminated in the Lorentz Ether Theory which was a perfectly consistent theory and explained all the facts. In this theory, the real speed of light is only c in that fixed stationary medium and time is absolute and universal and distances are absolute. It's just that our physical clocks run slower and our physical rulers shorten when we travel through that medium. In LET these effects are no different than other things that affect the measurement of time and distances, such as temperature, humidity, or pressure, for example.

    In LET and all other theories being proposed at the time, the principle of relativity (that all the laws of physics remain the same for all inertial observers) and the principle of the constancy of light (that the speed of a light ray is the same for all inertial observers) could not both be true. Every other theory had to compromise one or the other of these two principles.

    But Einstein was able to see that this conflict between these two principles (which weren't his ideas, by the way) was only apparent and not real. He started by raising them from principles to postulates, that is, assuming them to both be true at the same time without proof. Then he figured out where that would lead him and where it led him was that time and space were not absolute. He developed his time synchronization convention for distant clocks which allowed him, within his Theory of Special Relativity, to establish a consistent means by which to establish a system of time and distances which is called a Frame of Reference. This idea was brand new because now time is given a definition along with distances that are linked together rather than thinking of a universal absolute time and absolute distances.

    So within the context of a FoR in SR, we have a consistent way to address the one-way speed of light that allows us to make measurements of all times and distances and using the Lorentz Transform to reassign those measurements to another FoR moving with respect to the first one.
  5. Jul 30, 2011 #4


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    In order to measure velocity, you need two clocks, and a way to compare them. In order to compare them, you need a synchronization convention.

    There are some things you can measure with one clock, one has been given the name "celerity", see for instance http://arxiv.org/abs/physics/0608040. The term "celerity" remains a bit obscure - but since it's been published and we want a different name for the concept, I'll use it. Just don't be surprised if you mention it and people look a bit blank, if you're online you can refer them to the above paper for an extended discusion.

    Basically, you use an onboard clock to measure your travel time over a course of known distance, and that's a mesurement of celerity.

    Celerity is the same as velocity in Newtonian mechanics, but it's different, due to time dilation, in relativistic physics.

    So, does this get us anywhere with measuring the one way speed of light? Well, no, because while we can use celerity to measure the speed of any physical object, (as long as we can find a rugged enough clock that doesn't degrade its accuracy when it moves), we can't use it to measure the speed of light, ,because you can't put an onboard clock on a light beam.

    But there is a very important issue that we should mention, and that's the issue of isotropy, which is one of the fundamental principles of relativity.

    I think the cleanest exposition of isotropy is that the ratio of celerity to velocity should be the same in all directions. We know that velocity and celerity are the same in Newtonian physics - we can't keep them the same in relativistic physics, but we can insist that the ratio be independent of direction.

    I haven't seen anyone actually use this specific definition, generally people just talk about isotropy without actually defining what they're talking about, which makes the whole very important topic a bit muddled, in my opinion. It's much better to make the concept specific, by making it measurable.

    So the important question becomes "what clock synchronization is isotropic"?

    The answer to that is that, according to relativity, Einstein clock synchronization is the unique clock synchronization scheme that's isotropic.

    So, to recap, you can have lots and lots of clock synchronizations, but only one is isotropic.

    One other issue is that Newton's laws are most definitely NOT invariant if you change clock synchronization.

    You'l find that momentum is not equal to mv in the Newtonian limit if you use anything other than an isotropic (Einsteinian) clock synchronization. This is something that's also not stressed, and I think it should be.

    It's also easy to show why momentum can't be equal to mv with arbitrary clock synchronziation. Using an isotropic clock synchronization scheme, collide two objects of identical mass, one moving east, the other west, at an equal velocity v. They collide, and stop.

    Change your clock synchronization, and repeat. The measured velocity has now changed, due to the change in clock synchronization the velocity in one direction isn't the same as the velocity in the other.

    Let me expand on this. The trip takes 10 seconds with the isotropic synchronziation. We change the clock synchronziation such that the trip takes 9 seconds for one object, and 11 for the other. The velocity is now (d/9 sec) for one object, and (d/11 seconds) for the other.

    But the masses still collide, and stop. This is only possible if momentum is not mv, the expression for momentum must also be non-isotropic if the clock synchronization is non-isotropic.

    This is another way of operationally defining isotropy - for a while it was my favorite, but it uses the physical property of momentum as part of the defintion, and the definition based on celerity does not need to include such extraneous concepts to define it.
    Last edited: Jul 30, 2011
  6. Jul 31, 2011 #5
    Thanks very guys for the replies, it was very much appreciated.

    I had a read through the paper prevect quoted here: (thanks)
    I got the gist of it, but the math is a bit beyond me at the moment.

    I also found a link to this paper: http://arxiv.org/abs/gr-qc/0409105" [Broken]

    That was cited in an old thread here:

    I also found a Stanford lecture on YouTube here:
    Which I found quite useful. I still don't get the math fully but understood enough to see how the rationale behind how the Lorentz Transformation was derived.

    Where all this has really helped is I now have a much better understanding of how SR came about, particularly from the point of view of the math and how the Lorentz calculations were driven by Einstein’s postulates so that the speed of light was the same in all FoR and how it all links together. I know that may sound trivial, but I don't think I fully understood that before!

    As regards the one-way speed of light, I can now see how the choice of convention will affect the outcome. And although there is no direct way of testing the one-way speed of light, there does seem to be lots of empirical evidence to support the rationale behind Einstein’s clock synchronization.

    So doesn’t all this make the need for having empirical evidence for the velocity of the one-way speed of light almost unnecessary?

    Anyway, there does seem to be something dissatisfying and almost paradoxical about a theory derived from the speed of light being the same for all FoR that doesn’t allow one to test it empirically. (From a one-way speed of light point of view I mean.)

    So I was thinking, if we can’t measure the one-way speed of light by means of a natural convention, but we do know that two-way speed of light is the same in all FoR, couldn’t we say that we don’t need to test the velocity of the one-way speed of light and we just need to know that there is not a difference in the speeds of the two directions?

    Then shouldn't we be able to test for that, because if there was a difference in speed due to direction then that should be visible in all FoR?

    EDIT: Also we wouldn't need synchronised clocks to test for a difference in speed due to direction.
    Last edited by a moderator: May 5, 2017
  7. Jul 31, 2011 #6


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    I think your thinking is similar to mine - that the real issue is be isotropy, that the whole issue of the "one way speed of light" is mostly a dead end. If you know you have isotropy, then you know that if you choose coordinates that respect this isotropy, a fundamental symmetry of space-time, you'll have the speed of light the same in both directions.

    However, I'm not entirely clear on how you plan to experimentally test for isotropy - for instance, if you're using my idea of comparing celerity to velocity, or have some other experimental protocol in mind.

    This mirrors my complaint about the literature in general - it seems to realize that isotropy is important,but somehow the issue of how to define and test it experimentally never comes up :-(.
  8. Jul 31, 2011 #7
    Yes, it is a key concept, and it actually is covered in some treatments. See, for example, this online article:

  9. Jul 31, 2011 #8
    I had an idea, but it seemed too simple. If there is one thing I've learnt so far is then when it seems simple it usually means I don't understand something more fundamental! :redface:

    I also don’t know it falls down when it is Lorentz transformed into another FoR as I haven't learnt to do the calcs yet.

    Anyway, I've put my thought experiment below to test for isotropy. See what you think.

    Take two mirrors M1 and M2 and separate at a known distance of x. In the middle is a light source that will be split in two and send two beams of light in opposite directions. Each beam of light will travel at a slight angle towards the two mirrors at either side.

    The light source is then reflected back from the mirrors but instead of meeting in the middle, it continues until it hits to opposite mirror. (As it is at a slight angle.)

    The experiment is set up so each split light beam travels 1.5 units of length, 0.5 units in the first leg and 1 unit on the second leg.

    When a beam of light hits a mirror, it will start a timer which in connected to the mirror. When the opposite beam of light hits the same mirror, it will stop the timer. This will obviously happen on both sides.

    To check for isotropy, as each light source will travel the same total unit of length but a different unit of length in a particular direction, if there is any preferred direction, it would show on the timers.

    If the timers read the same, then the light beams are isotropic.

    If there is a difference, to test for errors in the system, I can just turn the apparatus 180 degrees but leave the light source the same.

    If I get the same error, but on the opposite side this time, it shows the error was in the system and that the light was isotropic.

    Here is a pic to help explain.

    http://img593.imageshack.us/img593/1892/lightbeamtest1.jpg [Broken]

    As I said, I don’t know how a moving observer would see the test. However I didn’t know if that mattered as I am not measuring the velocity of the light.

    I thought what would be important is that each observer measures the delay between the first light beam hitting mirror 1 (path AB) and the second light beam hitting mirror 1 (path EF) and vise versa for the other mirror.

    So even though not all observers would agree on simultaneity, (ie when the beam terminate on the respective mirrors, I though that they should agree on any difference (or not) in the timers. E.g. Interval 1 and Interval 2 as shown above.
    Last edited by a moderator: May 5, 2017
  10. Jul 31, 2011 #9
    Hi Samshorn, thanks for the link.

    For a layman like me, does this mean that the isotropy of light has already been demonstrated and that Einstein’s clock synchronisation is the right convention to ensure that fact? (And this should be the 'natural' convention to use.)
  11. Jul 31, 2011 #10
    Yes, the isotropy of light speed (in vacuum) in terms of inertial coordinate systems (in which "the equations of mechanics hold good", i.e., in which mechanical inertia is homogeneous and isotropic) is an empirical fact.

    Well, Galileo's synchronization, based on inertial isotropy, and the Poincare/Einstein synchronization, based on light-speed isotropy, are one and the same synchronization. This is why light speed is isotropic in terms of inertial coordinate systems. (This shouldn't be surprising, once we recognize that light energy has inertia - E=mc^2.) The "convention" consists simply of our free choice to describe and quantify space and time intervals in terms of inertial coordinate systems. We are free to use other coordinate systems, in terms of which neither light-speed nor mechanical inertia are isotropic, but those are usually regarded as unnatural coordinate systems.

    Yes, most people regard the inertial coordinate systems conceived by Galileo, Newton, and Einstein to be the most natural, i.e., to most accurately represent what we mean intuitively by space and time intervals. The meaning and definition of such coordinate systems hasn't changed since Galileo. All the was changed by special relativity was our understanding of how relatively moving systems of inertial coordinates are related to each other (i.e., by Lorentz transformations instead of by Galilean transformations).
  12. Jul 31, 2011 #11
    Thanks for that.

    Just one more question if I may, why all the fuss then to find the one-way velocity of light? It's not necessary from what I understand.
  13. Jul 31, 2011 #12


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    Amongst experienced relativists, there is no fuss at all. They understand that the reason the one-way speed equals the two-way speed is because we define it that way, and that's all there is to say.

    However, there are some people who fail to grasp this, and are convinced we ought to be able to do an experiment to discover what the "real" one-way speed is, and fail to understand that we can make the one-way speed almost anything we want it to be by synchronising clocks in a different way.
  14. Jul 31, 2011 #13
    It depends on what you mean. If you're interested in knowing the one-way speed of light in terms of any standard inertial coordinate system, then that is easily measurable, and is well known. In fact, it's easy to find the one-way speed of light in terms of ANY defined system of space and time coordinates.

    The problem is that some people think it ought to be possible to determine the speed of light without specifying any coordinate system. They have a metaphysical notion of "speed", a notion that is - in their minds - prior to any defined quantification of spatial and temporal intervals. Alas, it isn't possible to measure a metaphysical notion.

    Usually when this is explained to people, they say "oh, yes, now I see, my question was based on faulty thinking". Unfortunately, some people persist to the end of their days, seeking a way to measure their metaphysical notions. They will say things like "the speed in terms of inertial coordinates is merely the apparent speed, not the true speed". But if you ask them to explain what they mean by "true" speed (if it's not speed in terms of some well-defined system of coordinates), they fall back on metaphysical statements, insisting that true speed can be conceived even in the absence of any definite quantification of spatial and temporal intervals. Or else they say they believe Lorentz invariance will be found to fail at some point - which is what it would take for their metaphysical notions to become physical. But no violation of local Lorentz invariance has ever been detected. It's the same with inventors of perpetual motion machines - they believe that violation of energy conservation will be discovered, vindicating their belief in perpetual motion machines. It's a point of faith, not amenable to rational modification.
  15. Jul 31, 2011 #14


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    If I am understanding your point correctly then yes, that is testable, that is essentially what the Michelson Morely experiment tested. It (and its more accurate successors) found that the two-way speed of light is isotropic.

    I know that you are a little concerned about the "untestability" of the one-way speed of light, maybe this will help.

    It turns out that the fact that the one-way speed of light is not testable is a consequence of the simple idea that nature doesn't care what coordinates you use to describe physics. The result of any physical experiment does not depend on if you are using spherical coordinates, or Cartesian coordinates, or the Einstein synchronization convention, or some other convention. It is all just arbitrary labelling of coordinates.

    The fancy term for this is "diffeomorphism covariance":
  16. Aug 1, 2011 #15
    I think (hope!) I understand that now. In fact it is seems obvious once it sunk in.

    However, if I am being totally honest, I am still struggling with a couple of things around the relationship between choice of coordinate system or convention and the laws of nature.

    So if you will forgive my ramblings…

    I would have thought we want to choose a coordinate system that is consistent with our observations of the natural world.

    For example, I could think up some completely weird coordinate geometry to define the motion of planets in a such a way that makes the speeds of planets further away from the sun go faster. So in effect all plants orbit the sun in the same time.

    Obviously, anyone with a good telescope would see that mercury orbits the sun more times than Venus for example. So the defined ‘speed’ of the planets does not match the observations and these observations should be consistent for all observers.

    All of which is just a long winded way of saying that there must be coordinate independent observations we can make, well at least independent in part maybe, I guess all observations would require the time element.

    I saw the isotropy of light to be similar. Although the one-way speed is defined though our choice of convention, if there was some fundamental law of nature which meant the speed of light was not isotropic, we should be able to observe this independent of our choice of convention, as it is not ‘speed’ I am trying to observe, just difference.

    Now I know even ‘difference’ is frame dependant in some cases, but not all.

    So I thought of the experiment I proposed above. I assumed that this could demonstrate that light is indeed isotropic independent of any spatial coordinate system or convention, (except time) as the test does not require any specified convention, in the same way mercury orbiting the sun more times than venus doesn’t.

    Does any of that make sense or am I still getting it all worng?
  17. Aug 1, 2011 #16


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    You could certainly do that, but perhaps a more understandable example would be simply using spherical coordinates. The universe doesn't require you to use Cartesian or spherical coordinates. You can use either or both, depending on convenience and personal whim. The numbers are simply labeling conventions for us to keep track of things, not physical things that we find out in nature.

    The principle of diffeomorphism covariance says that any measurement from any given physical experiment must be independent of coordinates. What different coordinate systems will disagree on is the "meaning" of a given measurement, not the actual measured value. E.g. one coordinate system may say that a measurement is the length of some object, another may disagree that the measurement is the length, but it will agree on the number that the measurement produced.
  18. Aug 2, 2011 #17
    The only absolute consistency requirement is for there to be a one-to-one correspondence between a set of coordinates and the points or events being labeled. Given that this requirement is met, you can't arrive at any inconsistency with correct reasoning. For convenience we also usually impose the diffeomorphic requirement, meaning that neighboring coordinates correspond to neighboring points or events (even though this is somewhat problematic).

    Not true. Mercury and Venus actually do each complete one orbit per year... but this doesn't contradict any observations, because a year for Mercury is shorter than a year for Venus. The point is, you can't arrive at any contradictions by valid reasoning from any system of coordinates that satisfies the basic one-to-one mapping requirement. (You also need the diffeomorphic requirement to enable differentiation so you can even define things like speed in the usual way.)

    No, if you could establish absolute isotropy, you could trivially establish absolute synchronization. Simply use your isotropic signals from the mid-point between two clocks to synchronize them. But absolute isotropy is a meaningless concept, like one hand clapping. Measurement inherently consists of comparing something to something else (e.g., comparing a ruler to the edge of your computer to measure its length). It is possible to compare the isotropy of light speed with the isotropy of mechanical inertia, and this shows that the speed of light is indeed isotropic when expressed in terms of standard inertial coordinates. This is an empirical fact. But we can't call this absolute isotropy, it is just inertial isotropy. (Just as we can't say the Sun is the absolute center of the solar system, but it is (nearly) the inertial center.) The principle of inertia just happens to be the most powerful and successful organizing principle ever conceived, and it also happens (not coincidentally) to correspond to our intuitive notions of time and space.

    Still wrong. Your proposed experiment would give the same elapsed intervals on both clocks, regardless of whether the speed was isotropic or not. Think about it. If the signals were isotropic, let's say they each take 10 seconds to go from the center to the edge, and 20 seconds to go from one edge to the other. In this case we would expect both clocks to show an elapsed time of 20 seconds. Now suppose the signal is a bit faster in the rightward direction, so it takes only 9 seconds to go from middle to edge, and 18 seconds to go from edge to edge (in the rightward direction), but the leftward speed is unchanged. The left hand clock will start at t=10 and the right hand clock will start at t=9. The left clock will stop at 9+20 = 29, and the right clock will stop at 10+18 = 28. So the elapsed time on the Left is 29-10 = 19, and the elapsed time on the Right is 28-9 = 19. Same elapsed time on both clocks.
  19. Aug 2, 2011 #18
    Hi rede96, I promised to summarise and add to my earlier explanations in a new thread, but as you already started one I'll just place it here, with some addition related to your a) and b).
    I Note that much of it has already been said by others here - that's the advantage of a dedicated thread. So I'll first answer your questions, then give a - possibly boringly long - introduction of the light postulate with some subtleties, and then I'll explain my answers. :wink:

    Note that according to international standards the speed of light is c by definition; but I'll reply following the old definition, else thinking is blocked. :yuck:

    a): Quite so: you measure in reality the (average) two-way speed. However, you can verify that the definition leads to consistent results.

    b): It depends on what you mean with "different directions"!

    As Poincare explained, the simplest laws are obtained by choosing the one-way speed of light equal to the two-way speed of light. Probably based on that, in Einstein's 1905 paper,
    http://www.fourmilab.ch/etexts/einstein/specrel/www/ ,
    a standard reference system is set up in such a way that the one-way speed becomes equal to the two-way speed of light due to the Poincare-Einstein synchronization convention. As Einstein emphasised, this so "by definition".

    Thus the one-way speed of light as defined with respect to a standard reference system is equal to the two-way speed of light, which was assumed to be "a universal constant—the velocity of light in empty space". Einstein raised that assumption to a postulate.

    He formulated is as follows in 1907:
    "We [...] assume that the clocks [of a certain reference system] can be adjusted in such a way that the propagation velocity of every light ray in vacuum - measured by means of these clocks - becomes everywhere equal to a universal constant c, provided that the coordinate system is not accelerated."

    Obviously one has the free choice to adjust the clocks differently, in which case one makes the one-way speed of light different from c; the light postulate isn't affected by such a choice. Thus Einstein did not postulate that the one-way speed of light is isotropic [edit: in opposite directions (see further)]; instead he defined it to be so. Next he used that definition for a formulation of the light postulate; and he also formulated the same postulate without that definition.

    In SR, physical concepts such as "speed" are purely operationally defined; the definitions have no ontological meaning. *

    Now, returning to your questions:

    Concerning a): we can verify that for example light from a xenon bulb has the same speed as light from a light match. If this were not so, the definition would have to be refined for a specific light source. Another consistency check I'll discuss concerning b). And all such checks would also test the light postulate.

    Concerning b): as the one-way speed can simply be made equal to the two-way speed of light by definition (if one likes!), the verifiable (or "physical") content of the second postulate is that the two-way speed is the same everywhere in all directions (isotropic), and independent of the motion of the source.
    This can be tested: one aspect was already tested before by Michelson-Morley, who found that the two-way speed was the same in all directions (up to the limit of precision). This highlights that the synchronisation convention merely defines isotropy of the speed of light in each arm of the interferometer.
    Obviously, if you make the one-way speed equal to the two-way speed, then in principle you can test the same issues with the one-way speed as with the two-way speed. :smile:

    Note that with GRT the intended meaning of "universal constant" turned out to be partly incorrect; consequently the field of application of SR was reduced, just as the field of application of Newtonian mechanics had been reduced with SR. Einstein also explained that in his popular 1920 book:
    - http://www.bartleby.com/173/22.html

    *Compare the definition of electric current in a wire; while it was arbitrarily assumed that invisible electrons in the wire flow from plus to minus (which was wrong), the definition that current in a wire flows from plus to minus has been maintained.
    Last edited: Aug 2, 2011
  20. Aug 2, 2011 #19
    Could you expand on this a little please. I'm not quite sure what you mean.

    Yes, of course. I think I understand this. I supposed I could specify some sort of gravitational time dilation effect that would say that they both orbit ‘equally’, and make up some math to predict this special effect.

    But thought it would be difficult for that convention to be consistent for all observations. So the point I was making is that if I observe the two planets, seeing mercury orbiting more times than Venus, I would say they are not equal. Hence I would want a convention that demonstrates this, which is also consistent with other observations that I see as ‘not equal’.

    I almost got this! I can see what you are saying about absolute isotropy but I need to think it through for my benefit. However I am still struggling with the concept that every observation is a 'measurement' and if that matters anyway.

    Ah, you spotted my deliberate mistake! o:) You are right of course.
  21. Aug 2, 2011 #20
    Hi Harrylin, thanks for your post.

    I think this is beginning to sink in now, although I do find it difficult at times to get my mind out of the Stone Age! I wish they would have taught this stuff at school!

    Although the logic to all this is now apparent and I can see the error of my ways :smile:, there is just something unsatisfying about being able to 'define' nature in arbitrary and different ways.

    I am from the school that for any experiment we do in nature there can only be one outcome. I know quantum physics has changed this to some extent, but for the macro world, if I drop a ball for example, the possibilities are that it can move in a uniquely defined direction or it will stay where it is. It won't do both or move in different directions simultaneously.

    Similarly if I drop two balls there is a set of unique possibilities of which one set will be the outcome. Not a combination of two or more.

    So despite totally agreeing with the logic of all this, I find it difficult to accept that whether the two balls fall together or one faster than the other is just a matter of choice.

    Thanks for that, but I think I'll stick to SR for now! :smile:
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