Titlebook: Lecture Notes on the General Theory of Relativity; From Newton’s Attrac Øyvind Grøn Textbook 20091st edition Springer-Verlag New York 2009

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Covariant Differentiation,We must have a method of differentiation that maintains the anti-symmetry, thus making sure that what we end up with after differentiation is still a form.

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Curvature,The covariant directional derivative of a vector field . along a vector . was defined and interpreted geometrically in Sect. 5.2, as follows.

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,Einstein’s Field Equations,Energy–momentum conservation for a Newtonian fluid in terms of the divergence of the energy momentum tensor can be shown as follows.

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The Schwarzschild Spacetime,This is a solution of the vacuum field equations . for a static spherically symmetric spacetime. One can then . the following form of the line element (employing units so that c=1):

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Black Holes,Surface gravity is denoted by . and is defined by where . is the horizon radius, . for the Schwarzschild spacetime, . is the time component of the 4-velocity. The 4-velocity of a free particle instantaneously at rest in the Schwarzschild spacetime: The only component of the 4-acceleration different from zero is .

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,Newton’s Law of Universal Gravitation,ties, i.e. velocities much smaller than the velocity of light and “weak” fields. Weak fields are fields in which the gravitational potential energy of a test particle is very small compared to its rest mass energy. (Note that here one is interested only in the absolute values of the above quantities

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Cosmology,s reference particles. Then we introduce a “comoving coordinate system” in this frame of reference with spatial coordinates χ, θ φ. We use time measured on standard clocks carried by the galactic clusters as coordinate time (cosmic time).