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Geometric Understanding of Tensors.

  1. Jan 21, 2012 #1


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    I am a beginner in theory of GR and am trying to understand it better.

    I have a problem with understanding tensors. I got the algebriac idea, incliding covariance, contravariance and transformations etc of tensors. But not the geometric. Tensors are abstract but can I not have geometric interpretation of Tensors like I can have of Vectors.
    If I can get image of tensors like I have of vectors, I think they would be far easier.
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  3. Jan 22, 2012 #2


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    No guarantee but you can try to imagine tensor as transformation of coordinate system (just picking new coordinate axes). That's for covariant tensors. And contravariant tensor as transformation of coordinate units.
  4. Jan 22, 2012 #3
    I have been wondering about this myself; just like in your case I totally get the algebra and analysis behind it, but struggling to find a geometrical interpretation. In the special case of a contravariant rank 2 tensor in three dimensions I can sort of visualize the tensor as collection of three vectors, like e.g. in the mechanical stress tensor, representing forces in three spatial directions. When such a tensor acts on a vector, it transforms it into a new vector, like if a physical force had acted on it. This of course isn't mathematically rigorous, but it does help to understand the concept a bit better. Problem is, this doesn't really work for higher-order tensors, or tensors with mixed components.
  5. Jan 22, 2012 #4
    I know they discuss it in "Gravitation" by Misner, Thorne & Wheeler.
  6. Jan 22, 2012 #5
    yeah, tell me about it!! and, maybe unfortunately, you'll find there is a lot more mathematics to understand...Considering there are only limited exact solutions, I think, to the ten Einstein equations, making physical interpretations is not obvious....if it were, when Einstein developed the equations, he would have found solutions himself and science would not have argued about their meaning for a decade or more. Keep in mind Einstein somehow intuitively understood the physical nature of gravity, and with help from friends found the mathematics to fit....he did NOT derive his fundamental understanding from mathematics to the physcial....nor did anytone else at that time.

    If you are studying via formal schooling, taking the time and effort to learn the math
    is likely desirable and necessary. If its for hobby and self learning, be prepared for a
    long effort. Check some threads here and see the difficulty 'experts' have in reaching specific interpretational agreements and communicating them regarding different aspects of the mathematics.

    I only recently discovered in these foums that there is not even an agreed upon definition for a gravitational field...sure it's a 'curvature', but exactly how do you measure it?? There is no single metric [measurement]that takes precedent!!!!

    Wikipedia summarizes solutions to the Einstein Field Equations like this:

    See here and decide what you think:


    Also here's an interesting perspective from Einstein online:

    "In part, gravity is an observer artefact: it can be made to vanish by going into free fall. Most of the gravity that we experience here on earth when we see objects falling to the ground is of this type, which we might call "relative gravity". The remainder of gravity, "intrinsic gravity", if you will, manifests itself in tidal forces, and is associated with a specific property of geometry: The curvature of spacetime."

    Last edited: Jan 22, 2012
  7. Jan 22, 2012 #6
  8. Jan 22, 2012 #7


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