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How do you interpret the Robertson-Walker metric

  1. May 19, 2006 #1
    The Robertson-Walker (RW) metric is, in a spatially flat Standard Model universe, a simplemodification of the Minkowskian metric of Special Relativity (SR), thus:

    The SR metric of space-time along any arbitrarily chosen line, where a coordinate r measures distance in metres from some arbitrary origin, is :

    ds^2 = c^2 dt^2 - dr^2.

    (Here t is time and s is the interval between events. ^2 means squared and d indicates differential coordinate separation)

    For a flat Standard Model universe the RW metric may be obtained by inserting a scale factor a(t) as a multiplying metric coefficient for the (now re-gauged and assumed to be co-moving with matter) coordinate r, thus:

    ds^2 = c^2 dt^2 - a^2(t) dr^2 .

    The scale factor, which is a function of (universal) time, measures the homogeneous expansion of the highly symmetric SM universe, in which along any line proper distances in metres are a(t) r.

    The effect of this modification on the propagation of light is to introduce a red-shift of light transmitted between any two points of space-time, when a(t) increases monotonically with time.

    The essence of this modification is that it effects a change in the ratio between the metric coefficients of the time and space coordinates. The spatial metric coefficient is selected to be the one that is to be re-gauged and to differ from the SR value of unity. Because this is the choice that is made (rather than letting the time coefficient, or both coefficients, differ from unity via appropriate factors) we call the change "expansion".

    My question is:

    are such choices arbitrary, and is the use of the term "expansion" to describe change in cosmology therefore only a semantic convenience?
  2. jcsd
  3. May 19, 2006 #2


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    The R-W metric on its own is just mathematics, to turn it into physics you have to ask: "How do I measure length and time? What rulers and clocks do I use?"

    If you use a steel rule and an atomic clock, then you are defining as constant: fundamental particle masses, the charge of an electron and Planck's 'constant'. This means the diameter of the atoms in a ruler and the frequency of an atomic clock will not change as the particle is translated across space and time.

    Under this assumption of 'fixed rulers' and 'regular clocks' the dx's and dt's of the metric relate to physical distances and times as measured by comparison with those rulers and clocks. In addition mass is measured by comparison with the standard kilogramme.

    In this case the universe is one of fixed rulers around which space expands, the observed red shift may be described as a 'cosmological doppler effect'. Note I say cosmological doppler because the observed objects are not moving through space as with ordinary classical and (SR) relativistic doppler but it is the 'fabric' of space that is itself expanding with the passage of cosmological time and these objects are being carried along with it.

    Other alternative theories allow the atomic constants to vary, in which case some other standard has to be used as the basis of measurement of mass, length and time. The trick is devising such an alternative theory that is truly consistent and concordant with observations.

    Last edited: May 19, 2006
  4. May 19, 2006 #3
    I remember having read that one can also choose the same scale factor (i.e. a(t)) as a multiplier for the term c^2dt^2 in which case the time coordinate t is called as conformal time.
  5. May 20, 2006 #4


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    I think the simple question is that no, such choices are not arbitrary---on the contrary they are natural choices following from the usual practice of maintaining constant standard (using a single standard of distance)

    Any other choice would, I suppose, be described as arbitrary.

    but this choice is as natural and UNarbitrary as assuming that the wavelength of light from (a certain transition in) hydrogen is the same in Andromeda galaxy as it is in ours.

    that the speed of light is the same in Virgo cluster as it is in our local group of galaxies.

    In other words (along lines of what Garth said) you just assume the natural thing, that basic physical objects---atoms wavelengths speed of light, standard masses etc. ----are the same everywhere.

    so you measure distance in the most straightforward way, and that gives a universe that IN THE LARGE is approximated by FRW metric.

    the real universe has some bumps and dimples due to concentration of matter, but if you smooth all that out ------AND measure in a straightforward way by standard measures----you get a match to FRW metric.

    there are tricky variations with fancy coordinate systems which do NOT correspond to real clocks and rulers and uniform measures of distance. they might be useful for some purposes like a logarithmic plot of data can be handy, but they distort, like a mercator map shows Greenland too big---so I'd call those other systems arbitrary. Probably to some philosopher ANYTHING you choose is arbitrary----but I am making a naive intuitive distinction between the natural obvious choice and an artificial one that departs from usual conventions.


    the other question the answer would be No, expansion is not "just a semantic convenience"
    it has a kind of stark obviousness that smacks you in the face when you observe the sky in COLOR (or more exactly in different wavelengths)

    another interesting effect of the expansion is that out beyond a certain distance---the farther away objects appear LARGER: if two galaxies are in reality the same size then the further one will fill a larger angle in the sky

    that is because you are looking back into an earlier time when galaxies were crowded closer together, and each one was larger in proportion to the space between them

    it is a fun effect. we had part of a thread about it a few weeks ago, which maybe I can find.
    Last edited: May 20, 2006
  6. May 20, 2006 #5
    Semantics again

    In this case the universe is one of fixed rulers around which space expands, the observed red shift may be described as a 'cosmological doppler effect'. Note I say cosmological doppler because the observed objects are not moving through space as with ordinary classical and (SR) relativistic doppler but it is the 'fabric' of space that is itself expanding with the passage of cosmological time and these objects are being carried along with it.


    I become wary when folk adopt the common practice of talking about "space" or "the fabric of space" expanding.

    The meaning of these terms is somewhat moot. Einstein himself seems to have been not entirely clear on how to define or describe empty space. In The Meaning of Relativity he wrote rather vaguely of Poincaré's views, theorems of congruence, and forming a space of reference by "continuing" one body until it met another — thus giving a kind of operational definition of space as something that can be filled with contiguous bodies. But material bodies, we now know, are after all mostly empty space.

    Then Margenau and Lindsay wrote that space is:"..an abstract construct possessing those properties of rigid bodies that are independent of their material content", a definition entirely unclear to me.

    I do prefer operational definitions. My own definition of space is "that in which you can swing a cat", but I'm interested in more precise definitions. Any offers?
  7. May 20, 2006 #6


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    One question is: "How are space and time connected between adjacent events?"

    This is the basis of geometry and the answer is given by the metric, in SR the Minkowski metric and in GR by a more general form with generally non-unitary metric components.

    These GR deviations from the Minkowski metric describe the curvature of space-time as described by the Riemannian tensor.

    In the spherically symmetric solution this curvature of 'empty' space-time describes the trajectories of geodesics that predict accurately the passage of planets and photons in the solar system. So we have experimental evidence to support this hypothesis.

    In the cosmological homogeneous and isotropic solution to Einsteins's field equation the metric turns out, not surprisingly to be rather simple and spatially symmetrical.

    However as time progresses the space dimensions either inflate (or deflate) as determined by the scale factor R(t).

    It is in this sense that I said that space itself is expanding and carrying everything embedded in it along with it.

    The next question as I posed in my post #2 above is trying to make sense of this statement by relating the mathematics to the physics.

    This is done through the equation of state and the application of a conservation principle, which in GR is the conservation of energy-momentum, but not, in general, energy.

    The conservation of the energy-momentum of a particle means its 'rest' mass is constant and therefore its size and atomic frequency are also constant. Thus atoms become the 'fixed' rulers and 'regular' clocks used to measure the universe, which expands around them.

    The measure of the expansion of space is the increasing distance between cosmological objects observed as a red shift.

    Thus if you define atomic size and frequencies as your ruler and clock, as in GR, then space expands around them according to its cosmological solution. This is consistent with observation.

    Last edited: May 20, 2006
  8. May 20, 2006 #7
    The standards we use

    Thanks, Garth, for your replies. I hope I have the protocol and computerese of quoting you correct this time.
  9. May 20, 2006 #8


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    Yes indeed I would hold that the fundamental measurement is one of time, all you need is a clock and radar!

    However, this depends on the assumption that the speed of light in vacuo is constant, and there are some alternatives, the VSL theories that allow this speed to vary.

    It then is a question of which fundamental constants you allow to vary and which remain constant. If everything varied we would never know what we are talking about!

  10. May 21, 2006 #9
    The curiousness of the RW metric

    In a question I asked in my original post, namely: "is the use of the term "expansion" to describe change in cosmology therefore only a semantic convenience?" (in the RW metric, which accommodates the red-shift by a change in the ratio of metric coefficients) you kindly

    It is of course it is true that you observe an increasing wavelength-shift in light from galaxies, all the way out to as near as you can see to the red CMB horizon of the observable universe. But there is only “a kind of stark obviousness that smacks you in the face” about its cause if you share the perspective of nineteenth century astronomers, who were steeped in the tradition of measuring stellar radial proper motions with the Doppler effect. Hubble of course shared this perspective.

    As far as I am aware there is no other observational evidence that mandates the choice of “expansion” to describe change. It is the great body of other evidence, largely to do with the CMB, that supports the Standard Model, together with the logical structure of the model
    itself that makes “expansion” so persuasive a choice.

    But we live in a new Millennium, and physics has changed since the early days of modern cosmology. It is now much stranger than it was a hundred years ago; we live in a milieu where hordes of string theorists insist that there are many dimensions, only some of which have unfolded to create our present situation. And some cosmologists take the ravings of these theorists quite seriously.

    In this context the curious property of the RW metric that I have drawn attention to is, well, rather curious.
  11. May 21, 2006 #10
    Thanks for the information

    Are you referring to Weyl's conformal rescaling of the metric? I find his approach difficult to understand -- In my post I meant to point out only that essential feature of the RW metric is that it incorporates a time-varying change in the ratio of metric coefficients. But thanks for the head's up.
  12. May 21, 2006 #11


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    Hello Old!

    Just as a random head-up (because you seem to like getting a heads up) are you aware that modern cosmologists normally do not interpret the redshift as a Doppler effect. Somewhere in your post it suggest you think they they of it as a doppler effect. (?) but maybe you dont.

    what I enjoy is CONVERSING rather than arguing and you sound like you might be similarly inclined. If we get to arguing I will just stop, just so you know.

    Yeah, I like what you did by REMINDING about the orginal question you asked.

    I actually would say NO (as you pointed out) I dont think it is arbitrary or just a semantic convenience. I will try to EXPLAIN to you how I see it, if you are curious. I think expansion of space corresponds to a real physical process (there is even a process at fundamental planck scale that has come up in one or more QG theories which corresponds to it, but these theories are new and untested so bear caution in mind).

    In my view it is a real physical process, not an convenient arbitrary convention, and something WE MAY SOMETIME UNDERSTAND at the fundamental QG (quantum gravity---i.e. quantum dynamics of spacetime and matter as they interact at micro level) level but that we DON'T NOW UNDERSTAND.

    So instead of discussing current attempt to explain at fundamental level, which would get us into uncertain ground, I will just try to say why I think the PHENOMENON is physically real.

    But bear in mind that I am NOT TRYING TO PERSUADE you. I am not interested in making a "debate case" of it. I will tell you why I think it is real if you happen to be interested why I think that. YOU DONT HAVE TO AGREE.

    Before I start, you said a some things I dont understand.
    **...(in the RW metric, which accommodates the red-shift by a change in the ratio of metric coefficients)

    In this context the curious property of the RW metric that I have drawn attention to is, well, rather curious...**

    I dont see where you have described a "curious property" of the metric.
    It seems like a reasonable metric to me. I expect space to always be expanding or contracting because our best theory of gravity so far (1915 GenRel) says to expect that. (there are static solutions but they are not very stable or robust---much more typically a simple generic solution to the GenRel equation will be some metric that expands or contracts.

    Personally I dont see solutions to the Friedmann equation (the cosmologists simple version of Einsteins 1915 equation) as "ACCOMODATING" redshift by anything artificial. We know that metrics, to be solutions, will typically be expanding or contracting and one of the metrics which is an expanding solution happens to MATCH a number of observations.

    (not merely observed redshifts of galaxies but also the existence and features of the CMB, the increase in angular diameter with distance after a certain point)

    So I see things different from you on this score.

    the metric is a solution of a differential equation (Friedmann) which is a simplified version of the GenRel equation (Einstein) and the timevarying coefficient a(t) that you see in the metric is a solution to an equation WHICH DOESNT KNOW ABOUT THE REDSHIFT or the CMB data. the metric, with that a(t), comes out of solving the dif. eq. and then you check and it MATCHES. there are a few numbers you get to play with to make the fit good which you can think of as initial conditions for running the model. (you can adjust their present values or past values)

    for sure it is not ideal, there are big unanswered questions like what does Lambda stand for and why is it that size and why cant we see a lot of the matter we infer is there. Cosmologists love to talk about these huge mysteries and even dramatize them a bit.
    (but that said, they still have a very workable dif. eq. model with a small set of parameters to adjust and an impressive fit to a whole slew of various kinds of observations)

    I dont want to talk more about this----I am just explaining that I have a different attitude (the timevarying function a(t) in the metric is not something they play with to explain the redshift away-----essentially it has to be there and has to increase (or decrease) because that is how gravity works and we test our theory of gravity in the SOLAR SYSTEM)

    The reason I dont want to bother more explaining my attitudinal difference, is that I want to address the main question of HOW DO WE KNOW SPACE ACTUALLY PHYSICALLY REALLY EXPANDS?

    I will take a new post to do that.
  13. May 21, 2006 #12


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    what does "space expands" mean? what real thing does it refer to?

    Hello again Old!
    this is my main post to you.

    the operational meaning depends on being willing to imagine someone in a distant galaxy making the same measurements that we do
    (to understand me you have to be able to grant that distant galaxies are real places, not just painted on a stage backdrop, and pretty much like our galaxy in basic ways)

    the operational meaning of spatial metric expansion depends on giving operational meaning to the idea of being STATIONARY with respect to CMB.

    you look around and if the temperature is approx the same in all directions then you are stationary. if it is not the same then you adjust the data so they are from the standpoint of someone who is not moving

    fortunately not much adjustment is needed----our propermotion speed is only around 1/1000 of speed of light, and fortunately OTHER GALAXIES are not moving very much either as far as we can tell they TOO are approx stationary with very small propermotion drift speeds.


    we are almost stationary, and he is almost stationary-----for practical purposes we can forget about adjusting to make our platforms exactly stationary because they are already so close (and we still get a good firstorder picture of the situation)

    the operational meaning of spatial metric expansion is that the distance between two far-apart stationary points is always increasing


    Gravity in the solar system is well predicted by Gen Rel. Gen Rel is a theory governing spacetime metrics and also incidentally spatial metrics (which are spacetime metrics restricted to 3D so as to be spatial metrics). Gen Rel is the only good theory of gravity we have and it has passed repeated tests in SOLAR SYSTEM, and in EARTH ORBIT, and on earth SURFACE.

    Skepticism is good, I encourage you to doubt Gen Rel. But it has passed a lot of tests to high accuracy since 1915 and nobody has been able to think of better. WHEN THEY THINK OF BETTER IT WILL ALSO VERY LIKELY TELL US TO EXPECT SPATIAL METRIC EXPANSION.

    In other words, here is my personal view which you are welcome to doubt and contradict all you want, I say this:

    1. we have various ways of measuring distance (cepheids, H-R diagram, supernovas, moving cluster etc etc) that dont depend on redshift.

    2. we observe a general feature of the spatial metric is that distances between stationary points increase
    (which is operationally what expanding space means)

    3. this is a generic feature of spatial metrics that our theory of gravity independently tells us to expect.

    The best so-far theory of gravity says that one should not have the preconception that the distance between two stationary points remains the same. Only clueless people think that, because they see that is how spatial metrics behave approximately at short distance. But to get a workable theory of spacetime, so-far you need a theory that tells you to expect the normal thing to be that distances between stationary points should increase----or in other cases decrease----but anyway not stay the same. SPACE IS LIKE THAT.


    4. so far this is mostly a phenomenological discussion, but there is also some suggestion that when you try to construct a theory of quantum gravity you can find yourself confronted with models where at the fundamental level (dealing with what could be the fundamental degrees of freedom of spacetime geometry and matter as they interact) there turn out to be processes which seem to correspond at a fundamental level to the spatial metric expansion.

    and inverses of those processes corresponding to spatial metric contraction.

    but nobody I know is ready to have anything but a technical discussion of this among QG specialists, because it is too hypothetical. Maybe some people are talking about it in general audience context, I dont follow that closely so i dont know.

    I would say not to pay attention. But I just wanted to mention that I see those kinds of research directions and that it is POSSIBLE that one may eventually have some more fundamental description. But right now there is this kind of pragmatic effective description of the phenomenon.

    We have a theory of gravity. It works in the solar system and with binary pulsars etc etc. We get from it our basic understanding of what a SPATIAL METRIC should be. (because gravity is geometry and a theory of gravity is a theory of spacetime geometry). We expect spatial distances to be generally expanding or contracting.
    We set up the Friedmann simplification of the Einstein differential equation that is the law of how metrics behave----how geometry behaves. We solve it and get the RW, or the FRW as sometimes called.

    then we look out and see to our delight that it actually MATCHES.

    and finally, we return to the microscopic level and ask how matter is connected to spacetime geometry, and how both dynamically evolve, so that these wonderful things can be.
    Last edited: May 21, 2006
  14. May 21, 2006 #13


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    You could, of course, do a coordinate transformation on the FRW metric, but I don't think there exists any other coordinate system in which we can define a universal "time". If we want to give the entire universe a history, then I think we're stuck with the usual choice (with the scale factor), and in this coordinate system, "expansion" is an obvious description of the behavior. It is, however, just a colloquial description -- the true physical situation is more precisely described by the equations.
  15. May 22, 2006 #14
    Thanks for the long posts, Marcus. I'll absorb them in the fullness of time, as it were, and then converse about them.

    In the meantime .... I'm well aware that modern cosmologists don't think of the cosmological red-shift as a Doppler effect; GR offers a more sophisticated interpretation. But (and here's the rub) nineteenth and early twentieth century astronomers/cosmologists sure did, and the prejudice that this effect is associated with "expansion" (or, if you like, the proper motion of light emitting objects away from us) was in the first instance inherited from these savants, despite various later enlightening caveats (see Narlikar for example).

    The central suggestion I'm trying to make is that this this prejudice may be founded on a mathematical device that has no physical foundation. Or so it appears to me: in this conversation I'm trying to find out from others not what they believe (although this is quite fascinating), but whether there is a proper physical foundation for the seemingly universal acceptance of choosing the time-varying metric coefficient to be the one belonging to the space coordinate, namely the scale factor.

    Perhaps a careful reading of your and other posts will get me there.
  16. May 22, 2006 #15


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    The scale factor is derived from studying tensors in a maximally symmetric space, in particular cases where space-time is not maximally symmetric but space is, i.e. a foliation of space-time that satisfies the cosmological principle. If you constrain space to this degree of symmetry then all that can vary is a scale factor R(t) and the curvature k, which can take the values +1, 0, -1.

    Solutions for R(t) and k may be found, if in addition, Einstein's field equation is invoked and solved for such a maximally symmetric space. These solutions are dependent on the density parameter, equation of state and cosmological constant. As these are not known a family of solutions are produced and a precise description of the universe has to be determined by observation.

    The physical foundation for this interpretation is GR's success in accurately predicting outcomes of solar system experiments from the spherically symmetric solution of that same field equation.

    The cosmological solution, with its scale factor R(t), successfully predicted the expansion of the universe, so much so that Einstein originally could not believe this to be possible and himself failed to predict a priori Hubble Red shift when his own equations did so, they were 'cleverer' than he was!

    It further predicted the hot Big Bang phase with its hydrogen/helum abundance and the cosmological microwave background.

    The physical basis of cosmology, 'what is going on out there' is the experimental verification and interpretation of 'what goes on down here' in the laboratory.

    The interpretation of such a well tested theory is always open to further experimental testing, and possible falsification, such as is happening at present with the Gravity Probe B satellite experiment.

    Last edited: May 22, 2006
  17. May 22, 2006 #16
    Symmetry constraints on the RW metric

    Thanks, Garth. I think this may be what I am looking for. Have you a reference (not wikipedia, but a fuller treatment) for your last sentence that you can pass on to me?

    How do the VSL folk (for example Magueijo) cope with this constraint?
  18. May 22, 2006 #17


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    Any decent GR textbook such as Weinberg "Gravitation and Cosmology", 1972, chapter 13 "Symmetric spaces" page 375, which IMHO is the best. (See around equation 14.2.1 page 412).

    It is also touched on in Wald's "General relativity", chapter 5 "Homogeneous, isotropic cosmology", however Wald introduces the GR field equation too soon in my opinion.

    This can give the false impression that the R-W metric is derived from the GR field equation, it is not, it is derived from the Cosmological Principle alone.

    As I said above GR is then invoked, together with [itex]\Omega[/itex], an equation of state and [itex]\Lambda[/itex], to solve for R(t) and k.
    Although I think there are unsolved issues with VSL cosmology, this is not one of them, AFAIK Magueijo et al's theory fulfils the Cosmological Principle. The VSL aspect simply changes the function R(t) in the early phase of cosmic evolution as an alternative resolution of the horizon problem. (Note: if your interested, for another alternative resolution see: SCC)

    Last edited: May 22, 2006
  19. May 22, 2006 #18
    The scale factor and the CP

    Thanks for the references. I have found relevant material also in Bernard Schutz's Geometrical Methods of Mathematical Physics, section 5.19, which I have to hand. Others which aren't online will take longer to access.
  20. May 22, 2006 #19
    Semantics, after all?

    I agree with what you say.

    Just as it is better not to attribute the redshift to the classical Doppler effect, I think it would be better not to talk of expansion (especially of space or the fabric of space) or liken the universe to items on stretching elastic sheets or inflating balloons, even if this seems obvious at first glance.

    It is really not possible to describe some GR phenomena, such as these phenomena, or the interchange of time and space dimensions inside the Schwarzchild limit. The right words don't yet exist.
  21. May 22, 2006 #20
    I wonder, when galaxies reach the cosmological event horizon and accelerate and red shift to the point of disappearing, do they also accumulate mass, from our point of view, since they are speeding up so fast that we loose all contact with them? Some say that they are not moving relative to their space, so they don't increase in mass do to velocity. But we do see them red shift like a nearby object that is moving relative to space.
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