General relativity

Our exposition has brought together several theories of space, time, and motion. In pre-relativistic physics space and time are absolute, existing independently of the contents of the universe, and of each other. An instance of “now” is the same for every observer, at any point in the universe. Space has the manifold structure of ℝ³. Coordinates assigned to points xⁱ are arbitrary, but the distance between neighboring points, ( d s ) 2 = a i j d x i d x j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429023835/da3e512e-6dcd-4244-af51-23d185512f00/content/inline-math15_1.jpg"/> —where the metric a = diag ( 1 , 1 , 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429023835/da3e512e-6dcd-4244-af51-23d185512f00/content/inline-math15_2.jpg"/> , is independent of coordinate system and describes an intrinsic property of space. The Christoffel symbols vanish and the geodesic equation is d 2 x i / d t 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429023835/da3e512e-6dcd-4244-af51-23d185512f00/content/inline-math15_3.jpg"/> . Free particles follow straight lines at constant speed, x i = b i + v i t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429023835/da3e512e-6dcd-4244-af51-23d185512f00/content/inline-math15_4.jpg"/> (Newton's first law). The Riemann tensor vanishes, and geodesics do not deviate from each other. Einstein, in SR, modified the Newtonian framework by showing that space and time do not have a separate existence, but together form an absolute spacetime that has the manifold structure of ℝ⁴. Coordinates x^μ assigned to events are arbitrary, but the distance between events, ( d s ) 2 = η α β d x α d x β https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429023835/da3e512e-6dcd-4244-af51-23d185512f00/content/inline-math15_5.jpg"/> with η = diag ( − 1 , 1 , 1 , 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429023835/da3e512e-6dcd-4244-af51-23d185512f00/content/inline-math15_6.jpg"/> , is independent of observer and describes an intrinsic property of spacetime. The metric is constant, but with Lorentz signature. The Christoffel symbols vanish and geodesics are described by d 2 x μ / d λ 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429023835/da3e512e-6dcd-4244-af51-23d185512f00/content/inline-math15_7.jpg"/> . Worldlines of free particles are straight, x μ ( λ ) = a μ + b μ λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429023835/da3e512e-6dcd-4244-af51-23d185512f00/content/inline-math15_8.jpg"/> , a geometric feature that all inertial observers agree upon. The Riemann tensor vanishes, geodesics do not deviate, and we can establish a single coordinate system that covers all of spacetime.