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Vector and tensor fields
The spacetime of general relativity and paths of particles
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4-vector absolute derivative angle approximation axis basis vectors Cartesian coordinate system Cartesian coordinates components connection coefficients constant contravariant vector coordinate curves coordinate system covariant derivative covariant vector curved spacetime defined direction dot product dual basis DXa/du Einstein's energy equa equation of motion Euclidean space example Exercise expression field equations FIGURE flat spacetime fluid formula gauge geodesic equation gives gravitational field Hence implies integral line element Lorentz manifold mass massive object matrix metric tensor metric tensor field natural basis Newtonian notation null geodesics observer obtained orbit orthogonal parallel transport parallelly parameter path photon plane primed coordinate proper quantities r2 sin2 radial relativistic result rotating satisfy Show spacelike spatial special relativity speed spherical coordinates suffix surface symmetric tangent vector tensor field test particle theory timelike tion transported vector unit vector vector field wavelength zero