Fundamental Theories of Physics in Flat and Curved Space-Time Zdzislaw Musielak and John Fry Department of Physics The University of Texas at Arlington
OUTLINE General Relativity Our Main Goals Basic Principles Klein-Gordon Equation Application to Dark Matter New Physics at the Big Bang
From Special to General Relativity The Hole Argument (Einstein 1913-1915)
General Relativity Einstein (1915) developed General Relativity (GR) and established the following field equations G T where 8 G c 4 0,1,2,3 and 0,1,2, 3
Einstein s Tensor G where R R g 1 2 R g and are the Ricci and metric tensor, respectively R is given in terms of g and its derivatives and the Ricci scalar is R g R
Energy-Momentum Tensor For a perfect fluid T 2 ( p c ) u u pg is pressure and u is fluid s velocity p where is density, p The conservation of energy and momentum T 0
Approximate models of space-time
Mathematical Manifolds f : M x: U R m R ( f x 1 ) : x( U ) R
Mathematical Model of Space-Time A 4D continuous pseudo-riemannian manifold M endowed with a metric tensor which defines distance s between points on M 2 ds g dx g dx At any point P on M space-time is flat with the Minkowski metric. A freely falling object in M follows a time-like geodesic, which obeys where t t t 0 is a tangent vector
General Relativity and Its Limitations GR is our best theory of gravity and its predictions include: (a) Global structure and evolution of the Universe. (b) The Big Bang as the origin of the Universe. (c) Existence and structure of black holes. (d) Bending of light by gravitational fields. (e) Mercury s perihelion shift. Limitations of GR: (a) Describes only classical point particles and large masses. (b) Does not account for quantum effects. (c) Predicts singularities - limits on the validity of the theory.
Our Main Goals We do NOT want to quantize GR! We want to formulate a new fundamental wave theory in space-time whose properties are modified by the waves! We want to replace the field s equations of GR by new fundamental equations of the wave theory!
Principles of General Relativity 1. Principle of relativity 2. Principle of equivalence 3. Mach s principle Einstein (1918) The laws of Nature must be expressed by generally covariant equations. All coordinate systems are equivalent. Mach s principle forsaken (Einstein 1919, 1924).
Principle of General Covariance The universal laws of Nature must be expressed by equations that hold good for all coordinate systems, that is, are generally covariant with respect to arbitrary coordinate transformations Einstein (1916). Criticized by Kretschmann (1917) more than covariance Cartan (1923) covariant Newton s gravity Weinberg (1972) added another condition Torretti (1983) different versions of covariance and many others (Norton 1993; Requardt 2012). Einstein (1918, 1933, 1949, 1952).
Modern General Covariance Diffeomorphism is a mapping : M M, where is a f k 1 C mapping, and so is. : M R f ( f ), If and then ( M, g ) and ( M, g ) have physically identical properties gauge freedom! x x Diffeomorphism forms a group of general coordinate transformations Diff (M) - no physically acceptable irreps! GR is invariant under the transformations of Diff (M). Since the mathematicians have invaded the theory of relativity, I do not understand it anymore - Einsten
Our Basic Principles The Principle of Relativity / General Covariance: The laws of physics are the same for all observers, no matter what their state of motion. Equations of physics should have tensorial form. Local properties of curved space-time: The Minkowski metric, Poincare group and its irreps. Selecting a general class of observers: All observers are equivalent. The same physical object (elementary waves / fields) can be identified by all observers if they use tensors.
Klein-Gordon Equation Let be a scalar (complex) state function on a curved M with g (x). Let us introduce a vector field (x) q given by where q 2 i ( x) q ( x) ( x) 2 ( x) g ( x) q ( x) q ( x) 0 const with q (x) being real. Applying i and using 0, g we obtain [ g ( x) 2 0 ] ( x) 0 the Klein-Gordon equation in curved space-time given by g (x).
Applications to Dark Matter Most Dark Matter (DM) in galactic halo. How much DM in galactic disk? How much DM in solar stars, white dwarfs and neutron stars? KG equation in curved space-time will be applied to DM particles are free and ordinary matter gives the curvature. In-state is described by the Minkowski metric. Out-state is a curved space-time of stellar interior. Can DM be gravitationally trapped inside these stars? Can DM produce gravitationally induced radiation?
New Physics near the Big Bang Quantum effects are likely to dominate near the Big Bang singularity. All attempts to formulate quantum theory of gravity have failed! Formulate a new fundamental wave theory in which space-time is generated by the waves! Use the theory to investigate: (a) the origin of space-time, (b) the Big Bang singularity, (c) the Pre-Big Bang Era!
Things to Do! 1. Formulate a fundamental theory of elementary waves in (a) space-time whose properties are modified by the waves (equivalent to the GR equations but for the waves), (b) space-time that is generated by the waves. 2. Use the theory developed in 1(c) to investigate (a) the origin of space-time, (b) the Big Bang singularity, (c) the Pre-Big Bang Era. Supported by the Alexander von Humboldt Foundation.