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Thread: General Relativity Primer

  1. #1 General Relativity Primer 
    Administrator Markus Hanke's Avatar
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    MOTIVATION

    To give an overview of the main concepts relating to General Relativity. This is simply a quick reference, and not an exhaustive treatment of the subject; GR is a notoriously difficult subject, so a primer like this one cannot substitute for an in-depth textbook on the area.


    WHAT IS GENERAL RELATIVITY

    GR is a generalization of the physics in Minkowski space-time ( Special Relativity ) to space-times where the metric tensor is allowed to vary. In classical General Relativity this leads to the concept of space-time curvature. General Relativity is essentially a metric theory of macroscopic gravitation, and as such also forms the basis of modern models in cosmology.


    BASIC PRINCIPLES

    Equivalence Principle - This principle in its most basic form states that gravity and acceleration are equivalent. What that means is that an observer within an enclosed frame of reference cannot distinguish between gravitation and acceleration without reference to some other, outside frame.

    Space-time curvature - In GR the notion of gravitational forces is abandoned, and gravity is modeled as an effect of the underlying geometry of space-time itself. In other words, in GR there are no gravitational forces acting between bodies, it is rather space-time itself which possesses curvature in the vicinity of those bodies, and that curvature manifests itself in what we see as gravity. Here is a visualization ( click on image to enlarge ) :

    relativity_light_bending.jpg


    relativity_curved_space.jpg

    It needs to be made clear that curvature is a property of not just space, but space-time. Curvature in time manifests itself in the form of gravitational time dilation, meaning that clocks in the vicinity of massive bodies run slower as compared to a hypothetical observer at rest at infinity.
    Another consequence of curvature is that the shortest route between two arbitrary points is no longer a straight line, but rather a curved trajectory called a geodesic; this is the reason why uniformly moving particles will tend to curve towards a massive body - they are simply following the contours of a curved space-time.

    Covariance - What this means is that the laws of physics cannot depend on which system of coordinates is used to express them; in other words, the laws of GR are invariant under changes of the coordinate basis used to formulate them. This naturally leads to the use of tensors in the mathematics of GR.

    Gravitational Sources - In GR, all forms of energy are a source of the gravitational field, not just mass. That means that things like electromagnetic fields, stress and strain, electric charge, momentum and also the gravitational field itself all have an effect on gravity. This also means that the gravitational field is self-coupling and hence non-linear.


    MATHEMATICAL DESCRIPTION

    The mathematics of GR are notoriously difficult and tedious; essentially the theory is based on the formalism of Differential Geometry, which makes liberal use of objects called tensors. It is beyond the scope of this sticky to properly introduce and explain tensor calculus - for our purposes suffice it to say that the essential property of these objects is that they are covariant, i.e. independent of the particular choice of coordinates.

    The first and most basic piece of maths used in GR are the Einstein Field Equations :



    In the context of cosmology often a more general form including a so-called cosmological constant is used :



    The object is called the Einstein tensor, and is in essence a measure of space-time curvature; is called the Stress-Energy-Momentum tensor, and encapsulates information about the source of the gravitational field, i.e. matter, charge, momentum, stresses etc. is a proportionality constant, and is the cosmological constant.
    The Einstein tensor is a function of yet another fundamental object, the metric tensor , and its derivatives. It is the metric tensor which is a representation of the geometry of the underlying space-time, and its components are what the field equations are solved for. In standard GR the Einstein tensor is defined as



    wherein and are the Ricci tensor and the Ricci scalar, respectively. Both of these are functions of the metric tensor and its derivatives, and represent aspects of curvature.

    Putting all of this together we can see that the field equations are of the very general form “Curvature = Energy”, and that is exactly what the basic premise of GR is - that space-time curvature and the presence of energy go hand-in-hand.
    The other two elements of the mathematics of GR are the geodesic equation



    and the law of energy conservation




    GRAVITATIONAL WAVES

    One of the consequences of GR is the fact that the elements of the metric tensor can be periodic functions of time; such a solution describes a period oscillation in the curvature of space-time itself which propagates at the speed of light. This is called a gravitational wave. Such waves are the result of changes in the source of the field, e.g. two very massive bodies orbiting each other. Gravitational waves carry energy and momentum, much like electromagnetic waves.


    REFERENCES

    General relativity - Wikipedia, the free encyclopedia
    Prof T. Fliessbach : General Relativity, BI.-Wiss.-Verl. 1990
    Last edited by Markus Hanke; 01-27-2013 at 02:37 PM.
     

  2. #2  
    Administrator Markus Hanke's Avatar
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    Tests of Special Relativity & Relativity Principle

    Cryogenic Optical Resonators : http://www.exphy.uni-duesseldorf.de/...prl78_4741.pdf
    Non-Stationary Optical Cavities : http://www.exphy.uni-duesseldorf.de/...ingerLectNotes 2005-arxiv0510169.pdf
    Lorentz Invariance : Special relativity passes key test - physicsworld.com
    Time Dilation in Satellites : http://www.quantum.physik.uni-mainz...._861(2007).pdf
    Length Contraction in Heavy Ion Colliders : http://home.broadpark.no/~ccsernai/Csernai-textbook.pdf
    Relativistic Lorentz Force Tests : The effects of the Aharonov-Bohm type as tests of the relativistic interpretation of electrodynamics
    Anisotropy of Inertial Mass Tests : An Error Occurred Setting Your User Cookie
    Time dilation in mu-mesons : Measurement of the Relativistic Time Dilation Using
    Length contraction in free electron Lasers : What is SR, how is it generated and what are its properties?
    Length contraction in Penrose-Terrell Rotations : Can You See the Lorentz-Fitzgerald Contraction?
    Penning Traps : Antimatter tests of Lorentz violation - Wikipedia, the free encyclopedia

    Tests of General Relativity

    Universality of Gravitational Red Shift : http://www.exphy.uni-duesseldorf.de/...2-PRL10401.pdf
    Gravitational Potential at Short Distances : http://www.exphy.uni-duesseldorf.de/...2-PRL10401.pdf
    Tests of Lorentz Invariance : http://relativity.livingreviews.org/...005-5Color.pdf
    Gravitational Red Shift / Pound-Rebka : http://luth2.obspm.fr/IHP06/lectures...avRedshift.pdf
    Light Deflection within the Solar System/Shapiro Delay : [astro-ph/0302294] The Measurement of the Light Deflection from Jupiter: Experimental Results
    Lunar Laser Ranging to test Nordvedt Effect : Phys. Rev. 169, 1017 (1968): Equivalence Principle for Massive Bodies. II. Theory
    Hafele-Keating Experiment for Time Dilation : Around-the-World Atomic Clocks: Predicted Relativistic Time Gains
    Thirring-Lense Effect : http://www.nature.com/nature/journal...ture03007.html
    Geodetic Effect : Phys. Rev. Lett. 106, 221101 (2011): Gravity Probe B: Final Results of a Space Experiment to Test General Relativity
    Orbital Decay through Gravitational Waves in Binary Pulsar System PSR J-0737-3039 : Tests of General Relativity from Timing the Double Pulsar
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  3. #3  
    Administrator Markus Hanke's Avatar
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    DERIVATION OF GENERAL RELATIVITY

    Start with Newtonian gravity field, the validity of which is locally verified for weak fields :



    which is, expressed in terms of the energy-momentum tensor



    Now attempt a first ansatz to formulate this in a Lorentz-invariant manner :



    which further leads to a covariant formulation of the form



    Our task will now be to determine the unknown tensor G. We impose the following conditions on that tensor :

    1. G is a Riemann tensor
    2. G is composed of the first and second derivatives of the metric tensor
    3. The energy-momentum tensor obeys the usual symmetry and conservation laws and ; these properties then by default must also apply to our tensor G
    4. The theory must reduce to Newton's gravity for weak fields

    Using the above four points, the Bianchi identities, as well as the general ansatz



    plus a little tensor algebra, one find that the easiest tensor which satisfies all of the above conditions is

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