Pi and Space-time

  1. Is the value of Pi related to the curvature of space-time?

    Is it the value that it is because space-time is more or less flat?

    If the universe were of a greater open or closed curve, would Pi be a different value?

    Thanks,
    Glenn
     
  2. jcsd
  3. jcsd

    jcsd 2,226
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    Pi is a constant, so it's value is absolute. but if you take it as the ratio between the area of a circle and the square of it's radius then you are perfectly correct, it's value is dependent on the curvature of space time. If we imagine the universe (ignoring all other curvature) as negatively or postively curved then the ratio of a circle's area to it's radius squared would not by pi, but it would tend to pi the smaller the radius of the circle.
     
  4. I fundamentally believe that it's related to the curvature of space-time itself. If space-time had a different curvature, I believe that would have a different value. For our 3-dimensional space/time pairing, I suspect that 3.00000000000. . . would be the value if the cosmological constant were exactly one.

    Some current workers in this field suggest that the graviton must be a super-massive particle. I doubt this, however. Even if they find a super-massive particle that they believe to be the mediator of the gravitational force, I think that eventually a very much less massive particle will turn up. This is, of course, rather muddied by the fact that their is a relationship between mass, momentum and energy that makes mass a variable for any given object or moment. The fact that no quantizable unit of mass has been found subatomically, even more fundamental than quarks, may be due to this relationship.
     
  5. jcsd

    jcsd 2,226
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    Gold Member

    What's the cosmological constant got to do with this? I think you mean the density parameter [itex]\Omega[/itex] which is equal to 1 when the curvature of the universe [itex]\kappa[/tex] is equal to zero.

    Pi is not just a geometric property it is a number in it's own right and can only have 1 value and whatever the curvature of the universe this is not 3. Even in a non-flat space, the ratio of a circels area to it's radius squared can only equal three for certain sized circles in spaces with a certain curvature. In a truly flat space a circle will always have the ratio pi which is equal to 3.14.....

    I've never herad of the graviton evre being proposed as a massive particle as such a thing would go against what is known about gravity and what quantum field theory says.
     
  6. mathman

    mathman 6,521
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    Pi is the ratio of the circumference to the diameter of a circle in Euclidean (plane) geometry. It is a mathematical constant and has nothing to with the physical geometry of the universe.
     
  7. The carefully measured value of pi is determined by measuring the ratio of the circumference to the diameter of a circle in the real-world, where the curvature of space-time is non-zero! The carefully measured value of pi may be characteristic of the local curvature of space time. Likewise, phi is a constant that is characteristic of growth in the natural world, and phi is related to pi by
    phi = 2 cos(pi/5).
     
  8. HallsofIvy

    HallsofIvy 40,555
    Staff Emeritus
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    No, [itex]\pi[/itex] is not a "carefully measured value". It is an exact value based on abstract mathematics is not "measured" at all. [itex]\pi[/itex] does NOT depend upon the local curvature of space time. [itex]\pi[/itex] is the ratio of circumference to diameter in a (mathematically abstract) circle in (mathematically abstract) Euclidean space (which has measure 0). It makes no sense to talk about [itex]\pi[/itex] in that sense in a non-Euclidean space since then the ratio of the circumference of a circle to its diameter is not a constant at all.

    Yes, "phi is related to pi by phi = 2 cos(pi/5)", but phi is not "characteristic of growth in the natural world" except in a few special situations where we can find a number approximately equal to phi.
     
  9. No, pi would only be 3 if a circle were a hexagon.
     
  10. cristo

    cristo 8,411
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    This thread is extremely old (4 and a half years old, in fact!). I suspect that when the post was made, PF had a different level strictness than it does now. I think this thread should be closed or, at best, moved to the mathematics forum, since this has nothing to do with relativity!
     
  11. OK - found this (old) thread as I was looking for information on the relationship of pi and phi. This discussion though might be relevant to relativity, as space-time is considered to be Euclidean, with distortions due to mass. I accept the comment posted by HallsofIvy that pi is based on a mathematically abstract circle - but we live in the real world, with non-Euclidean space-time that we fail to notice. I.e. an empirically determined value of pi is based on a non-Euclidean circle. Some would argue that "all growth structures are regulated by the golden mean".
     
  12. D H

    Staff: Mentor

    We do not determine the value of pi empirically. It is not a measured constant, end of story. We already know the value of pi to far, far greater accuracy than the best any scientific measurement. The fine structure constant is one of the (if not the) best-measured physical constants; we know it to twelve places of accuracy. Many a nerd can spout pi to twenty places or more; with computers we know the value to millions of places.
    This thread doesn't have much to do with math, either, because pi is not a measured quantity. The best thing to do with this thread (and similar necromanced threads) is to nuke it to oblivion.
     
  13. Pi is totally mathematical constant and has no relation with experiment; it can be computed in many ways.
    Check this link for more information (http://en.wikipedia.org/wiki/Computing_π).

    So if there is a curvature of space then the circle will have different ratio (circumference/radius) than (2π), and that is one way to know that the space is curved, so the flat circle is from our (Mind Inventions and Creation) and have nothing to do with Physics, and Math is not an experimental science it is an absolutely (A Mind Creation).
     
  14. pi is a mathematical constant...if you introduce curvature of space-time into calculating the value of pi then you are introducing curvature of space-time into circle...so upon introducing this curvature,circle cant be a circle becasue the distance from the center to its boundary would be different in some places if you introduce curvature of space-time into circle and hence circle(curved spacetime) is not a circle and so the pi(curved spacetime) is not the pi...
     
  15. I kinda support Jeebus's comments. Here is why i do believe so. If you look at the way relativity was discovered, it goes like this

    1. Newton said - space-time and fundamental observables like mass are absolute.
    2. SR said - these are all relative
    3. GR said - space-time bends, they are not even straight.

    in all these, one thing is either assumed or taken as axiom - "Laws or nature are absolute, they will not vary". Laws of nature are essentially mathematics expressions.

    If you take realtivity to the next level you may get "Mathematics is not absolute either" - I know this goes against the current school of thought which was indeed started by Plato.

    What of Maths was relative. say - under heavy curvature of space time 1 electron + 1 more may not give 2 electorns. in other words let the number line be bent - this will distort PI, Exp and all other magical numbers. A straight numberline = euceldain geometry. wht if the number line was bent - bent against the imaginary axis ?
     
  16. I guess that would explain why the Bible has it as exactly 3.:rolleyes:
     
  17. but we also have to see at reality and not only at the abstract math
     
  18. D H

    Staff: Mentor

    This is getting ridiculous. [itex[\pi[/itex] is an abstract mathematical concept. We will not need to change the value for [itex]\pi[/itex] if we find the universe is curved. We already know that the circumference of a circle of radius [itex]r[/itex] on the surface of the Earth is (assuming a spherical Earth), [itex]c=2\pi R_e \sin\frac r {R_e}[/itex] rather than [itex]c=2\pi r[/itex]. This fact does not alter either the value of [itex]\pi[/itex] or the equation of the radius of a circle on a Euclidean plane.
     
  19. It is so Silly !!! Pi is totally mathematical constant, it will not be changed with any change in the space or time or atoms or electrons .............. If physics would try to change mathematics then all physics theories will be a trash, since that special relativity and the general relativity are based on (strong mathematical background + experiments) or else it will not be anymore an acceptable theory. So if you say that a physical theory will change mathematics, then you are saying that a theory will change itself.

    This issue is a philosophically problem, it belong to the (Philosophy of Mathematics).
     
  20. pervect

    pervect 7,952
    Staff Emeritus
    Science Advisor

    Yes, the thread is getting a bit silly. The point that Pi s a mathematical constant is absolutely correct. Sometimes people do not formulate their questions precisely, however, and the question of the ratio of the circumference of a circle to its diameter is also an interesting one.

    This ratio (of circumference to diameter) on a curved surface depends on the value of the radius. It always approaches Pi exactly as the radius approaches zero.

    http://en.wikipedia.org/w/index.php...198381761#2_dimensions:_Curvature_of_surfaces
    gives a formula for this ratio which can be deduced from the expression for Gaussian Curvature K:


    the point of which is that circumerence = pi * radius + terms of order o(r^3) for small r on a curved surface, and the coefficient of the term of order r^3 gives the value of the Gaussian curvature.
     
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