Brans-Dicke Scalar-Tensor Brans-Dicke Theory: . . . Theory of - PowerPoint PPT Presentation

Observable Time . . . How This Problem Is . . . General Relativity: . . . Motivations for . . . Brans-Dicke Scalar-Tensor Brans-Dicke Theory: . . . Theory of Gravitation At First Glance, the . . . Main Result: Cauchy . . . May Explain Let Us Use Gaussian . . . Proof: Idea Time Asymmetry Home Page of Physical Processes Title Page ◭◭ ◮◮ Olga Kosheleva and Vladik Kreinovich ◭ ◮ University of Texas at El Paso Page 1 of 15 El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit

Observable Time . . . How This Problem Is . . . 1. Observable Time Asymmetry: A Problem General Relativity: . . . • Most equations of fundamental physics are time sym- Motivations for . . . metric: Brans-Dicke Theory: . . . At First Glance, the . . . – starting from the ordinary differential equations (e.g., Main Result: Cauchy . . . the classical Newton’s equations of motion) Let Us Use Gaussian . . . – to partial differential equations describing physical Proof: Idea fields like electromagnetism or gravitation. Home Page • So, if we simply reverse the direction of time t , the Title Page resulting fields will satisfy the same diff. equations. ◭◭ ◮◮ • From this viewpoint, a time reversal of a physically rea- ◭ ◮ sonable process should also be physically reasonable. Page 2 of 15 • In practice, many physical processes are not reversible: Go Back – if we drop a fragile cup, it will break into pieces; Full Screen – however, a broken cup cannot get together to form a whole cup. Close Quit

Observable Time . . . How This Problem Is . . . 2. How This Problem Is Explained Now General Relativity: . . . • The problem of time asymmetry is known since Bolz- Motivations for . . . mann’s 19th century work on statistical physics. Brans-Dicke Theory: . . . At First Glance, the . . . • In modern physics, this problem is usually resolved by Main Result: Cauchy . . . assuming that the initial conditions are random . Let Us Use Gaussian . . . • Problem: this randomness assumption is outside the Proof: Idea usual PDE formulation of physical equations. Home Page • It is therefore desirable to come up with an alternative Title Page explanation within the PDE framework. ◭◭ ◮◮ • We show that the equations of scalar-tensor theories of ◭ ◮ gravitation are, in some sense, not T-symmetric. Page 3 of 15 • This may explain observed time asymmetry. Go Back Full Screen Close Quit

Observable Time . . . How This Problem Is . . . 3. General Relativity: Reminder General Relativity: . . . • In general, the field equations of a physical theory cor- Motivations for . . . respond to the minimum of the action Brans-Dicke Theory: . . . � L √− g dt dV, where g = det( g αβ ) . At First Glance, the . . . S = Main Result: Cauchy . . . Let Us Use Gaussian . . . • In particular, for the General Relativity theory (GRT): Proof: Idea 1 L GRT = 16 πGR + L mat , where Home Page Title Page • G is the gravitation constant, ◭◭ ◮◮ • L mat is the Lagrangian of matter, def = g αβ R αβ is the Ricci scalar, ◭ ◮ • R def = R γ Page 4 of 15 • R αβ αγβ , and • R δ αγβ is the curvature tensor. Go Back • Varying over g αβ , we get R αβ − 1 Full Screen 2 g αβ R = 8 πGT αβ , where T αβ is the matter’s energy-momentum tensor. Close Quit

Observable Time . . . How This Problem Is . . . 4. Motivations for Modifying General Relativity General Relativity: . . . • The observed gravitational acceleration a is often much Motivations for . . . larger that what follows from the observable mass M obs : Brans-Dicke Theory: . . . a ≫ GM obs At First Glance, the . . . . r 2 Main Result: Cauchy . . . Let Us Use Gaussian . . . • Traditional solution: there are non-observable masses (“dark matter”, “dark energy”). Proof: Idea Home Page • Problem: 95% of the mass is “dark matter” and “dark Title Page energy”. ◭◭ ◮◮ • Alternative idea: maybe the gravitational “constant” G is different at different locations, i.e., is a new field. ◭ ◮ • In such a theory, to describe gravitation, we need both Page 5 of 15 = 1 def the metric field g αβ and the new scalar field ϕ G . Go Back Full Screen • The corresponding scalar-tensor theory of gravitation was indeed proposed by Brans and Dicke. Close Quit

Observable Time . . . How This Problem Is . . . 5. Brans-Dicke Theory: Reminder General Relativity: . . . • In terms of this new field, the Einstein’s term 1 Motivations for . . . GR from Brans-Dicke Theory: . . . the Lagrangian takes the form ϕR . At First Glance, the . . . • We also need to add the effective energy density ϕ ,α ϕ ,α Main Result: Cauchy . . . ϕ Let Us Use Gaussian . . . of the scalar field, so we get: Proof: Idea R − ω · ϕ ,α ϕ ,α � � L BDT = ϕ + 16 πL mat . Home Page ϕ 2 Title Page • Varying over g αβ and ϕ , we get the following equations: ◭◭ ◮◮ R αβ − 1 2 g αβ R = 8 π � ϕ ,α ϕ ,β − 1 � ϕ T αβ + ω 2 g αβ ϕ ,γ ϕ ,γ ◭ ◮ + ϕ 2 Page 6 of 15 1 ϕ ( ϕ ; αβ − g αβ � ϕ ); Go Back 8 π Full Screen def � ϕ = ϕ ; α = T α ; α = 3 + 2 ωT, where T α . Close Quit

Observable Time . . . How This Problem Is . . . 6. At First Glance, the Brans-Dicke Theory is T- General Relativity: . . . Symmetric Motivations for . . . • At first glance, the Brans-Dicke Theory (BDT) is sim- Brans-Dicke Theory: . . . ilar to Einstein’s General Relativity: At First Glance, the . . . Main Result: Cauchy . . . – similar to General Relativity, the Brans-Dicke The- ory is described by 2nd order PDEs, and Let Us Use Gaussian . . . Proof: Idea – the BDT equations remain invariant if we reserve Home Page the order of time t , i.e., change t to − t . Title Page • In general, in a second-order theory, if on some Cauchy surface (e.g., for some moment of time t 0 ), ◭◭ ◮◮ – we know the values of g αβ , ϕ , and their first time ◭ ◮ derivatives ˙ g αβ and ˙ ϕ , Page 7 of 15 – then we can uniquely determine the second time Go Back derivatives ¨ g αβ and ¨ ϕ , Full Screen – and thus (at least locally) integrate the correspond- ing equations. Close Quit

Observable Time . . . How This Problem Is . . . 7. Main Result: Cauchy Problem for Brand-Dicker General Relativity: . . . Theory (BDT) Leads to T-Asymmetry Motivations for . . . • We show that if on some Cauchy surface, Brans-Dicke Theory: . . . At First Glance, the . . . – we know the values of the gravity tensor g αβ , its Main Result: Cauchy . . . first time derivative ˙ g αβ , and the field ϕ , Let Us Use Gaussian . . . – then we can determine ˙ ϕ from a quadratic equation. Proof: Idea • A quadratic equation, in general, has two solutions. Home Page • This means that in principle, for each initial condition, Title Page we can have two different dynamics. ◭◭ ◮◮ • In physical terms, our result means that BDT, in effect, ◭ ◮ consists of two T-asymmetric theories. Page 8 of 15 • The transformation t → − t transforms each of these Go Back two theories into another one. Full Screen • This T-asymmetry may explain the observed time asym- metry of physical phenomena. Close Quit

Observable Time . . . How This Problem Is . . . 8. Let Us Use Gaussian Normal Coordinates General Relativity: . . . • When g 00 = 1 and g 0 i = 0 for i = 1 , 2 , 3, BDT equa- Motivations for . . . tions take the form: Brans-Dicke Theory: . . . ϕ ) 2 � � At First Glance, the . . . − 1 i − 1 i = 8 π T 00 − 1 + ω + ω ( ˙ ϕ 2 + ¨ ϕ j κ j κ i 4 κ i 2 ˙ ϕ ; 3 + 2 ωT Main Result: Cauchy . . . ϕ Let Us Use Gaussian . . . 1 i ; j − 1 j ; i = 8 π ϕ T 0 i + ω ˙ ϕ 2 + ˙ ϕϕ ,i ϕ ,i 2 κ j 2 κ j ϕ ; Proof: Idea Home Page P ij − 1 κ ij − 1 4( κ ij κ k k − 2 κ k 2 ˙ i κ kj ) = Title Page � � ◭◭ ◮◮ 8 π T ij + 1 + ω + ωϕ ,i ϕ ,j + ϕ ; ij − κ ij ˙ ϕ ; 3 + 2 ωTγ ij ϕ 2 ϕ ϕ ◭ ◮ 8 π Page 9 of 15 ϕ − ∆ ϕ = ¨ 3 + 2 ωT. Go Back def def • Here, γ ij = − g ij , κ ij = − ˙ γ ij , P ij is the 3-D curvature Full Screen tensor, and all tensor operations are w.r.t. γ ij . Close Quit

Observable Time . . . How This Problem Is . . . 9. Proof: Idea General Relativity: . . . 8 π • From ¨ ϕ − ∆ ϕ = 3 + 2 ωT, we can explicitly express ¨ Motivations for . . . ϕ Brans-Dicke Theory: . . . in terms of γ ij , ˙ γ ij , and ϕ . At First Glance, the . . . • From the equation below, we can explicitly express ˙ κ ij Main Result: Cauchy . . . κ i (and, thus, ˙ i ) in terms of γ ij , ˙ γ ij , ϕ , and ˙ ϕ : Let Us Use Gaussian . . . P ij − 1 κ ij − 1 Proof: Idea 4( κ ij κ k k − 2 κ k 2 ˙ i κ kj ) = Home Page 8 π � T ij + 1 + ω � + ϕ ; ij − κ ij ˙ + ωϕ ,i ϕ ,j ϕ Title Page 3 + 2 ωTγ ij . ϕ 2 ϕ ϕ ◭◭ ◮◮ κ i • The resulting dependence of ˙ i on ˙ ϕ is linear. ◭ ◮ κ i • Substituting these expression for ˙ i and ¨ ϕ into the Page 10 of 15 equation below, we get a quadratic equation for ˙ ϕ : Go Back ϕ ) 2 − 1 i − 1 i = 8 π � T 00 − 1 + ω � + ω ( ˙ ϕ 2 + ¨ ϕ j κ j κ i 4 κ i Full Screen 2 ˙ 3 + 2 ωT ϕ. ϕ Close Quit