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    General Relativity 3: Astrophysics with Tensor Calculus
    General Relativity 3: Astrophysics with Tensor Calculus
    General Relativity 3: Astrophysics with Tensor Calculus
    Ebook64 pages49 minutes

    General Relativity 3: Astrophysics with Tensor Calculus

    By Robert Piccioni

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    About this ebook

    This book continues our exploration of the most profound theory of science, Einstein's General Theory of Relativity.

    All of modern astrophysics and cosmology stands on the foundation of General Relativity that is best expressed in tensors.

    This book, and its sequel, General Relativity 4: Astrophysics & Cosmology, present the clearest, most comprehensible, and most complete introduction to the tensor calculus of differential topology, which Einstein used to explain the cosmos. Derivations that are difficult to find elsewhere, are all collected here and explained in detail.

    This book presents all the principle tensors of General Relativity, and explains how these are computed and utilized.

    We apply our new expertise to:

    the derivation of the Schwarzchild metric
    the bending of starlight
    stable and unstable orbits around black holes
    the mass-energy tensor
    the precession of Mercury and binary pulsars.

    Those who love mathematical challenges need look no further. Others can bypass the appendices, which contain the most advanced material.

    LanguageEnglish
    PublisherRobert Piccioni
    Release dateNov 26, 2013
    ISBN9781310622458
    General Relativity 3: Astrophysics with Tensor Calculus
    Author

    Robert Piccioni

    Dr Robert Piccioni is a physicist, public speaker, educator and expert on cosmology and Einstein's theories. His "Everyone's Guide Series" e-books makes the frontiers of science accessible to all. With short books focused on specific topics, readers can easily mix and match, satisfying their individual interests. Each self-contained book tells its own story. The Series may be read in any order or combination. Robert has a B.S. in Physics from Caltech, a Ph.D. in High Energy Physics from Stanford University, was a faculty member at Harvard University and did research at the Stanford Linear Accelerator in Palo Alto, Calif. He has studied with and done research with numerous Nobel Laureates. At Caltech, one of his professors was Richard Feynman, one of the most famous physicists of the 20th century, and a good family friend. Dr. Piccioni has introduced cutting-edge science to numerous non-scientific audiences, including school children and civic groups. He was guest lecturer on a National Geographic/Lindblad cruise, and has given invited talks at Harvard, Caltech, UCLA, and Stanford University.

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      Book preview

      General Relativity 3 - Robert Piccioni

      Chapter 1

      General Relativity Basics

      To make this eBook more self-contained, this chapter reviews key concepts discussed in General Relativity 1 & General Relativity 2.

      SPACETIME

      Newton viewed space and time as unrelated, unchanging, absolute, and universal standards to which all measurements of location, motion, and rate of change were referenced. Any competent person measuring the length of a yardstick or the duration of a day would get the one and only right answer.

      Einstein said all of that was wrong. He said space and time are continually changing and are relative — different for different observers. The length of a yardstick and duration of a day have no single right answer. Each observer’s measurements are correct in their own reference frame, while being different from those of observers in other frames. Furthermore, Einstein said space and time are intimately related, two different aspects of one combined entity: spacetime.

      Newton viewed space and time like a Shakespearean stage, providing unchanging references for the positions and pace of a cosmic drama played out by the actors — objects with mass and energy.

      Einstein viewed spacetime like a Cirque du Soleil stage, continually changing, controlling actors’ motions, and playing an essential role in the cosmic drama. He showed that the laws of nature are best expressed in 4-dimensional spacetime.

      Each location in spacetime, each unique combination of t, x, y, and z, is called an event.

      To simplify the equations, we use natural units in which the speed of light and Newton’s gravitational constant equal 1: c=1 and G=1.

      CURVATURE OF SPACETIME

      The three types of curvatures — positive, zero, and negative — are illustrated below. Plane surfaces have zero curvature. Spheres curve inward in every direction, and are defined to have positive curvature (in this context, positive means greater than zero). That leaves negative curvature, such as the surfaces of a potato chip or a saddle, which curve inwards in one direction and outwards in another. The three curvature possibilities are shown below, along with their key geometric properties: the sum of the internal angles of triangles, and the ratio of circumference to diameter of circles.

      Three types of curvature.

      Note that a cylindrical surface is flat. General Relativity follows the nomenclature of topology by designating flat any surface that has zero curvature and is thus Euclidean. Imagine drawing a triangle and a circle on a flat sheet of paper and then rolling the paper into a cylinder. Rolling doesn’t stretch or tear the paper, nor does it change the angles of the triangle or the circumference or diameter of the circle. Hence Euclidean geometry applies equally on a cylindrical surface as it does on a plane surface — both have zero curvature and are flat.

      THE METRIC

      Everything we need to know about spacetime will come from something quite simple: knowing the distance between two points. The distances between all pairs of nearby points completely define the geometry of spacetime. An equation that provides those distances is called a metric, which forms the basis for all

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