What does it mean to be Covariant?

Discussion in 'Pseudoscience Archive' started by Green Destiny, Oct 17, 2010.

  1. Green Destiny Banned Banned

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    So, the best example I can think of is in fact, General Relativity, itself being a general covariant theory. But what makes a theory covariant? Exactly what does it mean?

    I looked up definitions:

    1. Physics Expressing, exhibiting, or relating to covariant theory.
    2. Statistics Varying with another variable quantity in a manner that leaves a specified relationship unchanged.

    Which is not very helpful, I am sure most agree. I wondered though if the latter statistical explanation is more akin to the works of a Covariant Theory. I do know that in general relativity diffeomorphism invariances in a schrodinger context can literally shuffle coordinates in spacetime but not alter the physical state. This seemed very similar to '' Varying with another variable quantity in a manner that leaves a specified relationship unchanged.''

    But perhaps I am barking up the wrong tree?

    Covariance by wiki-terms is much more similar in the way it explains it http://en.wikipedia.org/wiki/Covariance - so are my suspsicions right. Is this what makes a theory covariant or have covariance?
     
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  3. Acitnoids Registered Senior Member

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    I too have questions about this terminology but, I'm going to try a different tact. This is my first question. When I state that one electron Volt is equal to 1.602176487x10-19 joules and that one joule is equal to 6.241509647x10+18 electron Volts, are these two quantities inversely proportionate or covariant?
    1 / 1.602176487x10-19 = 6.241509647x10+18
    .
    How about this one. The elementary charge over plancks constant, in joules, equals 2.417989454x10+14 amps per joule (A J^-1). I know that this quantity shares the same quantitive value as one hertz because if you were to divide 2.417989454x10+14 into 1, as in 1/(e/h), it will equal the same numerical value as plancks constant in electron Volts. Does this make 2.417989454x10+14 A J^-1 covariant to 1Hz and h or are they unrelated to each other?
    .
    Lastly, one hertz is equal to 4.13566733x10-15 electron Volts per second, 299,792,458 meters and 4.7992374x10-11 Kelvin. I have always thought of these units as being unchanging specified relationships between various quantities (which sounds an awful lot like the O.P.'s second definition for covariance). Are these quantities covariant or are they nothing more than relationships between various units of measurement?
    .
    Like I said, I'm confused as to what terminology to use when describing these examples. To me the word "covariant" simply means equal but different. This definition is extremely vague which is why I'm having a hard time grasping the true meaning of the word. Any help will be appreciated. Examples are welcomed.
     
    Last edited: Oct 17, 2010
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  5. AlphaNumeric Fully ionized Registered Senior Member

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    The de Witt equation is not used in pure general relativity. The time evolution of say a cloud of dust collapsing into a black hole or the orbit of a planet around a star is nothing to do with the de Witt equation. And even when you do include quantum phenomena and consider the de Witt equation you don't automatically get Schrodinger equation phenomena. In fact even the Wikipedia page says that the Schrodinger equation isn't satisfied in some circumstances.

    Covariance is to do with the specific way a tensor transforms. It can be worded in different ways and can be taken to mean slightly different things depending on the context. A covariant derivative compared to a partial derivative include the connection so that it transforms properly under a coordinate transformation. A covariant vector transforms in the inverse way to a contravariant vector under a coordinate change, where one transforms via the Jacobian of the transformation, the other transforms with the inverse of the Jacobian. Thus a scalar formed by the contraction of covariant and contravariant vectors is unchanged under a coordinate transformation, as it should be.

    Didn't you cover coordinate transformations when you did vector calculus? Or did you miss that lesson along with the lesson on what 'curl' means?
     
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  7. Green Destiny Banned Banned

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    Aphanumeric, how can you say it's not pure generalized relativity?

    General Relativity is plauged by what is called the 'time problem of QM' and it's absolutely down to relativistic clocks and how time ceases to exist in General relativity. I am somewhat confused by your statements.

    The Wheeler de Witt equation is at the core of general relativity and is part of the frozen time problem inherent in the generalized solution of the wheeler de witt equation. I cannot see how it cannot be pure generalized relativity when it acts as a common solution to his equations. Hence, why Einstein often qouted these strange paradoxes of GR.

    Can you explain how it isn't pure relativity? The concept involves systems inside the universe and clocks which cease to tick.
     
  8. Green Destiny Banned Banned

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    Would anyone else like to give a stab at what constitutes covariance in a theory? Whilst alphanumerics answer is technically-correct, it's the same old which you read in textbooks and written mathematical explanations. It did not however tell me precisely the physical nature of the covariant nature.

    Again, I will ask. What makes a theory covariant? What is it, how is it different to a non-covariant theory? Is there any physical attrubute about them which can be said in a strightforward manner?
     
  9. Green Destiny Banned Banned

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    For instance, as AN said, it has to do with how tensors tranform. Remove this part, keeping in mind it's still part of it, but take into consideration a written work on covariant theories, and how they are manifestly part of a physical explanation?
     
  10. AlphaNumeric Fully ionized Registered Senior Member

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    Please explain how time ceases to exist in GR.

    Pure GR as in 'no quantum mechanics GR'.

    If you want a short, snappy and slightly sparing with the truth response then it means 'Independent of coordinate choice'. A partial derivative is not a tensorial object, a covariant one is. The connection \(\Gamma_{ab}^{c}\) is not a tensor but when you add it to the partial derivative you get a tensor. Or if you consider \(\Gamma_{ab}^{c}-\Gamma_{ba}^{c}\) then you get a tensor (torsion).

    People who read books and aren't wacko. Go and eat sand in the corner Terry.
     
  11. Acitnoids Registered Senior Member

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    I know this wasn't directed towards me but thank you none the less. There are still some things that I'm shaky on but at least I wasn't using improper terminology to describe what I thought I knew. In truth, I'm a little embarrassed. I have to switch between various coordinate systems almost daily (not at your level of transformations of course) yet I've never heard of the word "covariant" outside of this forum. It looks like I have some reading to do and I thank you for giving me a place to start (Jacobian, contravariant vector, contraction of covariant and contravariant vectors ...).
     
  12. BenTheMan Dr. of Physics, Prof. of Love Valued Senior Member

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    Thread cleaned.

    OP answered in Post 3, paragraph 2.
     
  13. kevinalm Registered Senior Member

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    I would suggest the book:

    A Brief on Tensor Analysis by J. Simmonds

    I'm currently reading it and it is really helping to clear up some of these points for me. There is even a pdf of it on the net, but I'm not certain as to copyright status so I won't post a link. It should be easy to find though if you google the title.

    Please Register or Log in to view the hidden image!

     
  14. Green Destiny Banned Banned

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    Thank you ben.
     
  15. Green Destiny Banned Banned

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    I will get you all the info you want. Give me some time.

    Thank you, for the wordy, yet short, explanation. That was what I was looking for.
     
  16. CptBork Valued Senior Member

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    An even simpler explanation without reference to tensors: a theory like General Relativity is covariant if all inertial frames are equivalent, i.e. no local measurements could distinguish one inertial frame from another and any inertial observer can legitimately claim they're the one at rest. This means that different observers might measure different quantities (for example, an observer moving with respect to a particle measures it to be heavier than an observer at rest relative to the same particle), but when making predictions based on those measurements, each observer uses the same set of equations, and does not need to worry about which inertial frame those equations might apply to.

    Edit: In fact, in a covariant theory you can write down the same equations in the same general form for any coordinate system you want, as long as the variables you plug into the equation are given with respect to that coordinate system.
     
    Last edited: Oct 19, 2010
  17. kurros Registered Senior Member

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    Should not one generally specify the group of transformations that ones tensors are covariant with respect to, i.e. 'Lorentz covariant' or 'generally covariant' or similar? Or does it actually not matter? I guess it is usually obvious from the context. I am a little uncomfortable thinking that the various tensors flying around in QFT can just be transformed any which way, but I guess it is quite possible that they can. I haven't done any curved space QFT so I am not sure what modifications one needs to make. I mean ok, if the tensors are contracted down to scalars then that seems like it's probably fine since the metric comes in there...
    hmm.
    I think it must matter since we could pick some really weird transformation group and make some tensors that only transform as tensors under that group...
    edit: actually maybe not, the wikipedia page on tensors has made me rethink my apparently flawed concept of tensors...
    edit: ok I guess the specification I am talking about really just amounts to specifying what space the tensors live in. If you define them on one manifold I guess it doesn't make sense to expect they would transform correctly if you pretend they lived on some other manifold...
     
    Last edited: Oct 19, 2010
  18. Green Destiny Banned Banned

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    Perfectly said. Thank you.
     
  19. Green Destiny Banned Banned

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    You did ask. And I have taken out some time to find some good reading materials.

    This is the best link available to get the ''just'' of why time ceases to exist as we know it. It has no flow. Nothing changes. Time ceases to tick.

    http://www.fqxi.org/community/essay/winners/2008.1

    As you will see, frozen time was a big topic for many of the authors.

    I recommend reading them all if you don't know of this amazing feature of relativity, and is a consequence of the Wheeler de Witt equation and is a solution of pure relativity, as you will see.
     
  20. Acitnoids Registered Senior Member

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    Thank you for the suggestion. I scanned the pdf whilst at work and, from what I could glean, it looked fairly informative to say the least. Amazon is selling it for less than forty dollars so I may go that route. I'm a little old fashioned. Most of the time I tend to prefer solidified pulp mash over electron driven photons

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    .
     
  21. kevinalm Registered Senior Member

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    You're very welcome. I came across the title while reading a some free lecture notes I downloaded. The author of those notes described the book as "the only book on tensor analysis I would recommend to anyone. Under any circumstances." or words to that effect. So far it seems to live up to that recommendation.

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  22. CptBork Valued Senior Member

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    I think the idea of covariance is that any object written in covariant form can be defined on some manifold (i.e. flat Minkowski spacetime Cartesian coordinates), and then if you know how that manifold maps to some other manifold under a change of coordinates, the rules of covariance automatically define how you write that object down in your new coordinate system. The change of coordinates could come from a boost, translation or rotation, a switch to a non-inertial frame, or it could even be something like a switch to polar coordinates where the physical situation remains identical.

    I don't see why you couldn't do this in quantum field theory as well- you just replace the derivatives with GR-style covariant derivatives (the two are equivalent in flat space), and that would tell you how your theory changes under general, smooth coordinate transforms. I guess the difference compared to GR is that GR ascribes certain physical properties to certain coordinate systems that might not have any inherent physical meaning in a conventional, flat space QFT.
     
    Last edited: Oct 21, 2010
  23. Green Destiny Banned Banned

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    That's a nice definition cptbork.
     

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