Fundamentals of Newtonian Mechanics3.1 Introduction3.2 Newton's Laws3.3 Frames of Reference3.4 The Equation of Motion for a Particle3.5 Conservation Theorems3.6 Conservation Theorems for a System of Particles3.7 Limitations of Newtonian MechanicsSuggested ReferencesProblemsChapter 4. Vector Calculus2.1 Introduction2.2 Differentiation of a Vector with Respect to a Scalar2.3 Examples of Derivatives -Velocity and Acceleration2.4 Angular Velocity2.5 The Gradient Operator2.6 The Divergence of a Vector2.7 The Curl of a Vector2.8 Some Additional Differential Vector Relations2.9 Integration of VectorsSuggested ReferencesProblemsChapter 3. Other points in the spacetime evolve over time as shown by the blue dashed lines, for which it is the negative φ solution that disappears-thus, there is no obvious way to choose, a priori, which root to follow if not the central one.Matrices and Vectors1.1 Introduction1.2 The Concept of a Scalar1.3 Coordinate Transformations1.4 Properties of Rotation Matrices1.5 Matrix Operations1.6 Further Definitions1.7 Geometrical Significance of Transformation Matrices1.8 Definitions of a Scalar and a Vector in Terms of Transformation Properties1.9 Elementary Scalar and Vector Operations1.10 The Scalar Product of Two Vectors1.11 The Vector Product of Two Vectors1.12 Unit VectorsSuggested ReferencesProblemsChapter 2. One could switch to the negative φ solution at this point, but this would create a discontinuity. Beyond this point, as the field continues to grow, there is no continuous real solution to the constraint.
The dot-dashed green lines show subsequent events, where the root has moved to the positive values at 2 and then to 3, where g ^ n n = 0 = d C E / d φ = 0. The solid pink line shows a general point at t = 0, when we choose D i E i = 0 and φ = 0 (point 1). Solutions exist for φ at the intersections with the x axis, where C E = 0. Each curve in this plot depicts the constraint C E at different times at a given point in space, to illustrate how solutions are lost when the ghost instability is triggered. The Proca evolution equation gives rise to a cubic constraint equation for φ of the form C E = a φ 3 + b φ + c = 0, with a = − μ 2 λ, b = μ 2 ( 1 + A i A i ), and c = D i E i. These instabilities have implications for the formation of condensates of massive, self-interacting vector bosons, the possibility of spin-one bosenovae, vector dark matter models, and effective models for interacting photons in a plasma. The system initially evolves as in the massive case, but instabilities are triggered in a finite time once the self-interaction becomes significant. We demonstrate that this is not the case by evolving a self-interacting Proca field on a Kerr background, where it grows due to the superradiant instability. It has been suggested that nonperturbative dynamics may drive systems away from such instabilities.
We show that this seemingly benign modification inevitably introduces ghost instabilities of the same kind as those recently identified for vector-tensor theories of modified gravity (but in this simpler, minimally coupled theory). Here, we investigate vector fields with self-interactions by replacing the mass term in the Proca equation with a general potential. Massive vector fields feature in several areas of particle physics, e.g., as carriers of weak interactions, dark matter candidates, or an effective description of photons in a plasma.
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