Mathematical Principles of Theoretical Physics
2.2 Lorentz Invariance With this principle, the special theory of relativity was developed in two directions: a) the introductio ...
48 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS wherecis the speed of light, andM^4 is endowed with the Riemannian metric: (2.2. ...
2.2. LORENTZ INVARIANCE 49 Theorem 2.18.The Minkowski metric (2.2.5) or (2.2.6) is invariant under the coordinate transformation ...
50 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS Lorentz Tensors We now define Lorentz tensors, also called 4-dimensional (4-D) t ...
2.2. LORENTZ INVARIANCE 51 3) The 4-D current density: (2.2.17) Jμ= (J 0 ,J 1 ,J 2 ,J 3 ), Jμ= (J^0 ,J^1 ,J^2 ,J^3 ) =gμ νJν, wh ...
52 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS Theorem 2.20(Lorentz Invariants). 1) The contractions (2.2.22) Aμ 1 ···μkBμ^1 ·· ...
2.2. LORENTZ INVARIANCE 53 Second, the 4-D acceleration is defined by aμ= duμ ds = (a^0 ,a^1 ,a^2 ,a^3 ), a^0 = 1 c √ 1 −v^2 /c^ ...
54 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS whereP= (P^1 ,P^2 ,P^3 )is as in (2.2.28). Thus, it follows that the relativisti ...
2.2. LORENTZ INVARIANCE 55 or equivalently, H 1 = ∂A 3 ∂x^2 − ∂A 2 ∂x^3 , H 2 = ∂A 1 ∂x^3 − ∂A 3 ∂x^1 , H 3 = ∂A 2 ∂x^1 − ∂A 1 ∂ ...
56 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS 2.2.6 Relativistic quantum mechanics Quantum physics is based on several fundame ...
2.2. LORENTZ INVARIANCE 57 and by (2.2.45), we derive from Postulate2.23an relativistic equation, called the Klein- Gordon (KG) ...
58 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS Then we have (2.2.57) E=cP±mc^2. Inserting the operatorsEandP 1 in (2.2.45) and ...
2.2. LORENTZ INVARIANCE 59 the left-hand side of (2.2.60) should be (2.2.62) ( iγμ ̃∂μ− mc h ̄ ) ψ ̃=R ( iγμ∂μ− mc h ̄ ) ψ, wher ...
60 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS By the expressions ofγμ, we infer from (2.2.66) that R=cosh θ 2 I−sinh θ 2 γ^0 γ ...
2.3. EINSTEIN’S THEORY OF GENERAL RELATIVITY 61 the space-time Riemannian space, general coordinate transformations, and genera ...
62 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS Consider a free particle moving in a Riemannian space{M,gμ ν}, which satisfies t ...
2.3. EINSTEIN’S THEORY OF GENERAL RELATIVITY 63 forces in (2.3.6). Hence we need to regard the Riemann metric{gμ ν}as the gravit ...
64 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS is continuous and non-degenerate. The inverse of (2.3.10) is denoted by (2.3.11) ...
2.3. EINSTEIN’S THEORY OF GENERAL RELATIVITY 65 and by (2.3.15) we have T ̃ijT ̃ij = tr [ (T ̃ij)(T ̃ij)T ] = tr [ (bkl)T(Tij)(b ...
66 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS To solve this problem, as shown in (2.1.31), we need to add a termΓto the deriva ...
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