# Introduction to tensor calculus for general relativity by Bertschinger E.

By Bertschinger E.

There arc 3 crucial principles underlying normal relativity (OR). the 1st is that house time might be defined as a curved, 4-dimensional mathematical constitution known as a pscudo Ricmannian manifold. briefly, time and area jointly contain a curved 4 dimensional non-Euclidean geometry. therefore, the practitioner of OR needs to be conversant in the basic geometrical houses of curved spacctimc. specifically, the legislation of physics has to be expressed in a sort that's legitimate independently of any coordinate procedure used to label issues in spacetimc.

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For Lagrangian L2 this is simple, with the result 1 H2 (pµ , xν , τ ) = g µν (x)pµ pν . 2 2 (8) Notice the consistency of the spacetime tensor component notation in equations (6)(8). The rules for placement of upper and lower indices automatically imply that the conjugate momentum must be a one-form and that the Hamiltonian is a scalar. The reader will notice that the Hamiltonian H2 exactly equals the Lagrangian L2 (eq. 5) when evaluated at a given point in phase space (p, x). However, in its meaning and use the Hamiltonian is very diﬀerent from the Lagrangian.

E. a tangent vector in the manifold). If we wish, we could make V a unit vector (provided V is non-null) by setting dλ = |dx · dx |1/2 to measure path length along the curve. However, we will impose no such restriction in general. Now, suppose that we have a scalar ﬁeld fX deﬁned along the curve. We deﬁne the derivative along the curve by a simple extension of equations (36) and (38) of the ﬁrst set of lecture notes: df ˜ V = V µ ∂µ f , ≡ ∇V f ≡ ∇f, dλ V = dx . e. the covariant derivative along V , the tangent vector to the curve x(λ).

However, it can be applied to any Hamiltonian system, relativistic or not. Next we write the integral invariant of Poincar´e-Cartan in terms of the new variables: ω = p0 dq 0 + pi dq i − Hdt = Pi dQi − KdT − d(Ht) + tdH . (26) Recall that this is a one-form deﬁned on M 2n+1 . Now let γ be an integral curve of the canonical equations (12) lying on the 2n­ dimensional surface H(p, q) = h in the (2n + 1)-dimensional extended phase space {p, q, t}. Thus, γ is a vortex line of the two-form pdq − Hdt = p0 dq 0 + pi dq i − Hdt.