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Schutz's Error - units of c

  1. Aug 21, 2014 #1

    On p.4 of "A First Course in General Relativity", Shutz says that we redefine the units of time so that the new unit of time is the meter, where one meter is "the time it takes time to travel one meter".

    He then argues that:

    ## c = (distance-light-travels-in-any-given-time-interval)/ (the-given -time-interval) = (1 m)/(the-time-it-takes light-to-travel-one-meter) = (1 m)/(1 m) = 1 ##
    I am afraid the above is wrong. Yes, you can rescale c so that you have 1 instead of ## 3X10^8##, but with respect to units, the correct derivation is:
    ## c = (distance-light-travels-in-any-given-time-interval)/ (the-given-time-interval) = (1 m)/(the-time-it-takes light-to-travel-one-meter) = (1 m)/(1 m/c) = c ##
    Whatever the *numerical* value of c is. But c is still measured in m/s.

    It’s somewhat like saying “we count the number of pumpkins using the unit of one cow. So from now on when you count pumpkins say “one cow, to cows…”, but keep in mind that “cow” means “pumpkin”.”
    Anyway, I will stick with Schutz for the time being, but personally the numerical value of c did not bother me and I don’t think anything good has been achieved by this rescaling of the numerical value of c. As for the units, it is simply wrong.

    If we redefine the units of time so that the new unit of time is the meter, where one meter is "the time it takes time to travel one meter" and apply this to the three space dimensions (x.y,z), this would lead to the conclusion that we have FOUR time dimensions (since the meter is a unit of time according to the way in which it was redefined above).
    Last edited: Aug 21, 2014
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  3. Aug 21, 2014 #2

    Simon Bridge

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    Using c=1 is massively useful in simplifying the equations.
    You'll discover that as you do more of these things.

    $$c=\frac{\text{distance light travel in an interval}}{\text{the given time interval}} = \frac{1\text{m}}{\text{the time it takes to travel 1m}}=\frac{1\text{m}}{1\text{m}}=1$$... is correct, and makes it clear why c has no units.

    Recall that the time it takes light to travel 1m is defined as 1m of time - exploiting the proposal that time is another kind of distance. You could say that time is being measured in units of meters-per-light if you like. (Which is pretty much what you did in fact...)

    Your version reads:
    $$c=\frac{1\text{m}}{(1\text{m})/c}\frac{1}{c}\implies c=c\implies 1 = 1$$ ... which doesn't get you anywhere.

    Many books do it the other way - choosing the second for time, and the light-second for distance ... i.e. "the distance that light travels in 1 second".

    It is not like exchanging pumpkins for cows because cows and pumpkins are different kinds of things while the second and the light-second are different words for the same kind of thing.

    We do not have to remember that the light-second is "so many" meters to use it, just like we don't need to remember how many inches there are in a meter to use meters.

    This is something you are going to have to get used to ;)
    Last edited: Aug 21, 2014
  4. Aug 21, 2014 #3


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    Or miles per second, or centimeters per second, or furlongs per fortnight.... You can use any units you please as long as you're consistent. There's nothing magic about m/s.

    If Schutz's logic bothers you, imagine for now that he's just saying that he's going to measure time in units of "greeples", and just as there are a thousand milliseconds in a second, there are ##3\times{10}^8## greeples in a second.
  5. Aug 21, 2014 #4

    My point exactly.

    But I'll stick with it; maybe it will make some sense later.

    And thank you for your reply.
  6. Aug 21, 2014 #5
    And another question related to this: is this simplification of the units of c what Einstein himself did or is it a simplification which became popular later on?

    Simplifications may be helpful but they may also obscure some of the meanings which can be derived out of a formula. But of course I haven't seen yet those complicated formulae you speak of.
  7. Aug 21, 2014 #6

    Simon Bridge

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    I know - it took me quite a while to get the hang of not seeing that "c" everywhere.

    Interesting question - making natural unit systems by setting c=1 (with some other constants) predates special relativity by some decades, and he was certainly aware of the utility since Plank units were current. Checking... he has used c=1 in The Foundation of the General Theory of Relativity (1916) (... see footnote p154); therefore the answer to your question is "yes" :)

    That's right - the simplification also brings out more fundamental relationships.
    Basically you'll end up expressing everything in terms of invarients and the geometrized units help with that, just like expressing energies in electron-volts simplifies E-mag (electromagnetism) when you are dealing with things on the scale of electrons and protons.

    Have you ever considered how arbitrary meters and seconds are?

    Basically, when we set c=1, we are letting Nature tell us what units to use rather than just making something up ourselves. This is pretty much guaranteed to lead to insights.

    See also:
    ... this is probably a clearer explaination of what is being done than your text;

    ... not great but a decent general overview of using natural units.
  8. Aug 21, 2014 #7
    Firs of all, thank you very much for your detailed response. (The system won't allow me to send you thanks; says I have given too many and I have to wait 24 hours...and I have been waiting for 24 hours since 2 days ago :p)

    Second, I have worked with units of electron-volt (basically took a course which covered more or less the content of "Introduction to Electromagnetism" by Griffiths or other similar books).

    And I did not find the units of electron-volt helpful AT ALL. If anything, I had to force my mind to "switch" to them because most authors use that unit...but that's all. It was a "necessary evil". I have read about the various "benefits" that some physicists claim this unit brings. And what I found out, at least in my case, is that the benefits would have been exactly the same if the original units (Joules) had been used. Yes, the numbers would have been different, but the relationship between numbers would have been absolutely the same. Nothing was made more "intuitive" (as some authors claimed), or "easier" through the use of electron-volt. In fact, it complicated things unnecessarily, at least in my opinion.

    The same goes for using Gauss instead of Tesla "because one Tesla is too big". Totally useless, in my opinion.

    Or, to depart a bit from Electromagnetism, let's talk about Quantum Mechanics, although the two certainly overlap. Planck's constant divided by 1second is way smaller than either the Joule or the electron-volt. So if you use electron volts instead of Joules, you just change the exponent, but the exponent (still a double-digit number in many cases) still remains. Again, at least in my opinion, totally useless. I know the charge of an electron is -1.6*10^-19 C, so I can compare the results in my mind if I want to. No need for electron volt. In fact, in order to obtain a grasp of the physical meanings, many times (in an *overwhelmingly* big number of times), in my mind, I have to do the opposite: convert from electron-volts to Joules and then see how the energy of this compares to the energy of that.

    So far the only, absolutely only, "alternative unit" which, I think, is useful is the light year, as it gives you *some* sense of the distances involved where the metre would be too small. But that is absolutely the only exception.

    And yes, I have considered how arbitrary everything is. I have pondered a lot about it actually.
    Last edited: Aug 21, 2014
  9. Aug 21, 2014 #8


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    Calculate the speed of light in lightyears/year.
  10. Aug 21, 2014 #9


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    That's going to turn out pretty useful, since SR says that what's "just time" or "just space" to one observer, is actually a mixture of space and time to another. A statement such as "these two events are separated in time but not in space" is never objectively true in SR.

    So having the same units for time and distance is going to turn out to be useful in the same way as it's useful to have the same units for "left-right distance" and "forward-backward distance". If you're facing north and a friend is facing northwest, it's not like one of you is objectively right about which point on the ground is 10 meters away in the forward direction. And if you want to calculate the distance to the point that your friend says is 10 meters away in the forward direction, things get really weird if you use different units, say lrmeters and fbmeters, for left-right distances and forward-back distances.
  11. Aug 21, 2014 #10

    Simon Bridge

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    Calculate the amount of energy gained by an electron being accelerated through 10V potential difference - once in electron-volts and again in Joules, then tell me how the Joules version has the same advantages.

    How about the rest-mass of an electron in kg vs in keV? Which uses the fewest characters to write out?

    Opinions are subjective - can you come up with an objective measure of what makes a unit more convenient?

    Sometimes I think it is a mistake to drill SI units into students as completely as we do.

    ... that would be the correct interpretation, provided "it" is light.
    There are indeed 4 time dimensions. It is equally accurate to say there are 4 space dimensions.
    That is exactly the insight that is made explicit - time and space are the same kind of thing.

    Certainly there are situations where SI units are valuable - or nobody would use them.
    You realize that feet and pounds have been used for many generations as well - do you have the same problems with those?

    Since you think that "1.6x1019C" is an easier number to work with than "1", I guess you'll just have to suffer on until the weight of calculations builds up.

    Gripping hand is: everyone but you (so far) finds them better so you are stuck with it.
  12. Aug 21, 2014 #11

    Thank you very much. Your explanation makes things a bit clearer.

    But in that case I think Schutz could have said "there is some equivalence between space and time which you will understand later", rather than "drop the heavy stone in the middle of the still lake" like he did.

    Seconds are still seconds, metres are still metres and c is still measured in metres per second BUT there is some equivalence between these units. Oxygen and plutonium are equivalent on some level (for example they are both made of protons, neutrons and electrons) but the former is needed by the human body whereas the latter is highly poisonous.

    On the other hand, if I taught about oxygen and plutonium to someone who has no idea what electrons, neutrons and protons are, *without* even mentioning what I wrote in the paragraph above, and if I asked him to "just accept" that oxygen and plutonium are "the same" or "almost the same" from a certain point of view, then the student would be highly confused.

    Schutz's introduction / explanation could have been MUCH clearer, even at beginner level.

    Thank you very much again Fredrik.
    Last edited: Aug 21, 2014
  13. Aug 21, 2014 #12
    Just for the sake of discussion:

    True; it is apparently easy in the case of such physical situations; the other side of the truth is that you end up having many people who just write automatically "eV" without bothering to *ponder* (as you said) what these units *really* mean.

    No; and, for the very reason you stated, the world cannot come up with an objective measure of what makes a unit more convenient for me (and I suspect I am not the only one)

    And it is also a mistake to drill eV (and other "convenient units" for that matter) into students as completely as it is done in most (all?) universities.

    Great answer, thank you very much :)

    As a matter of fact I do have a problem with them; I really do! (I am smiling as I write this; it is as if you "guessed my likes and dislikes" regarding units.)

    Yes, that seems to be the situation. Talking about drilling things into people's minds :p
    Last edited: Aug 21, 2014
  14. Aug 21, 2014 #13
    Another positive side of eV: it certainly looks better in a science magazine when you write: "we have accelerators that can accelerate electrons up to energies of 6.25 TeV " than when you write "we have accelerators that can accelerate electrons up to energies of 1 micro joule" :p
  15. Aug 21, 2014 #14

    Simon Bridge

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    That happens regardless of the units chosen.
    What does 1m "really" mean? You know, when I learned HS physics, 1m "really" meant the length of a lump of metal kept in a vault in France. 1 foot was the size of someones actual foot once upon a time.
    When you find something is x meters long, do you always think "oh that's x times the distance light travels in 1/299,792,458 of a second"? (The "real" meaning) or are you thinking about a length of wood you used in school?

    With the electron volt, you cannot use it without knowing that it is the energy gained by an electron accelerated through 1V.

    Curiously, it is possible ... we can do this by looking for things in Nature that are not a matter of opinion (so they are objective) and base our units on those.

    But what you are talking about is happiness... you'd be unhappy with any objective definition I suspect. Time should cure that. Mid you, you could always strive to be so important in science that you can get your own system of units accepted... then you'll be happy and loads of students can curse your name :)

    I don't think the alternate units get quite the dogmatic approach that SI units do.
    We used to get a rap on the knuckles for not using the "right" units.

    Well, when the switch to SI units was made, many people had much the same kinds of issues you do ... about SI units.
    They were resisted for many years in a lot of fields - the USA was notoriously non-SI for eg. so there are many engineering standards based on the inch. But there was even disagreement over "imperial" measures ... how big is a gallon?

    Do we use calories or kilojoules for the energy content of food?
    Do we use Curies or Baquerals or rads for radiation?

    The scientific answer is: "it depends".

    Anyway ... your question has been answered: the author did not make a mistake.
  16. Aug 21, 2014 #15


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    There are immeasurable advantages to natural units. You cannot claim the author is at fault just because you haven't yet seen the applications of natural units. I mean, would you claim that an author of an introductory mechanics book is at fault for choosing the units of force to be such that the proportionality between one unit of force and one unit of acceleration is just one unit of mass so that ##F = ma## instead of ##F = \alpha ma## for ##\alpha \neq 1##?

    This choice is just as arbitrary as ##c = G = \hbar = 1## but I don't see any complaints here regarding that.

    Anyways there are practical advantages to using Planckian units. Here is one example that is actually for reduced Planckian units. When you are numerically solving differential equations for scalar or tensor perturbations during inflation, you deal with numbers that are extremely, extremely large in SI units. Any finite step algorithm would take ages to solve differential equations which have coefficients separated in scale by entire orders of magnitude. Planckian units however makes all the relevant numbers small and manageable for the most part and makes numerical integration that much easier. Of course it's not always so nice since certain comoving wavenumbers will become quite large in magnitude at least during the start of inflation in Planckian units but it's a small price to pay.
  17. Aug 21, 2014 #16
    Now, after I have seen the explanations of those who posted in this thread, what I am saying is that the introduction was extremely poorly explained.

    Yes, I believe you when you say that later on I will see many applications of the equivalence of space and time.
  18. Aug 21, 2014 #17


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    You have to be a little careful here. As stated, this is not correct.

    There are 3 spatial dimensions and 1 temporal dimension. The number and nature of the dimensions can be determined from the signature of the metric, which is independent of the system of units and the coordinate system.

    What I think you meant is that you can measure time and space in the same units, e.g. 1 m of time or 3 s of distance. So you could have all 4 dimensions in units normally associated with time, and so forth.
  19. Aug 22, 2014 #18
    Yeah because physics from diag[1 1 1 1] is an awful lot different than from diag[-1 1 1 1] or diag[-1 -1 1 1].
  20. Aug 22, 2014 #19
    It actually makes things less obscure, if anything, to just set it at 1. You run all your speeds from 0 to 1.

    Thinking about c non-stop as being in m/s or perhaps feet per hour or miles per second or whatnot gets you distracted. Like thinking oh c is 186,000 miles per second or what might potentially give one the weird impression that there is something magical about 186,000 miles per second and that there is some weird reason that things are blocked at that number. And well how do we know something can't go 188,000 miles per second. I mean why not? What is so magical about 186,000 that things just get blocked.

    If you mix space and time and assume light always goes the same speed in all inertial frames and work it out you see that there is a natural range 0 to max otherwise you get into into square roots of negative numbers in situations where that would seem pretty weird. It becomes clear that it is a natural max, a limit. And it's not like thinking oh at so and so miles per hour things just get stopped for some who knows whatever reason that I'm not sure I should even believe in why some random number should just be the limit for speed.

    And it's makes thing easy to deal with if you normalize them all to a simple range like say 0 to 1.
    and you don't get distracted paying attention to junk all over or whatnot.
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