By Günter Ludyk
This ebook is an creation to the theories of distinct and common Relativity. the objective viewers are physicists, engineers and utilized scientists who're searching for an comprehensible creation to the subject - with out an excessive amount of new arithmetic. the elemental equations of Einstein's concept of designated and common Relativity are derived utilizing matrix calculus, with no assistance from tensors. this option makes the ebook targeted and a worthwhile device for scientists and engineers without event within the box of tensor calculus. partly I the principles of designated Relativity are constructed, half II describes the constitution and precept of basic Relativity. half III explains the Schwarzschild resolution of round physique gravity and examines the "Black gap" phenomenon. Any invaluable mathematical instruments are person pleasant supplied, both without delay within the textual content or within the appendices.
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Extra resources for Einstein in Matrix Form: Exact Derivation of the Theory of Special and General Relativity without Tensors
The above is 1 γ2 u2 − 2 u v v u c γ 2 (1 − c2 )2 1 (1 − v u 2 ) c2 1 2 1 u − 2 u v 2 γ c 2 − 2γ 2 u v 2 −2 u v 2 2 + γ 2v2 + v2 . 74) (without γ ) one gets u2= 1 (1 − v u 2 ) c2 u2 + −(u v)2 − v 2 u2 −2 u v c2 2 + v2 . 75) one obtains 1− u2 v 2 v 2 u2 u2 1 1 − = − 2+ 4 c2 c2 c c (1 − vc2u )2 = 1 (1 − v u 2 ) c2 1− u2 c2 1− v2 , c2 and this implies 1− u2 = c2 1− u2 v2 1 − c2 c2 1− v u , c2 or with 1 def γu = 1− u2 c2 and 1 def γu = 1− u2 c2 , we finally get γ 1− v u c2 = γu . 77) γu c . 79) now is transformed by a Lorentz transformation L into the velocity vector u : u = Lu.
61) dx dx vv dx = + (γv − 1) 2 + γv v dt dt v dt v u = u + (γv − 1) 2 v + γv v. 64) or, after addition of u − u = o in the numerator, w= v + u + ( γ1v − 1)(u − 1+ v u v) v2 v u c2 . 65) So this is the speed of the point P with respect to the reference system X . For a double vectorial product one has a × (b × c) = a c b − a b c. e. 65) one can also write w= v+u+ 1 1 ( v 2 γv − 1)(v × (u × v)) 1+ v u c2 . 67) it is clear that the sum of the two velocities is v+u w= . 65) it follows that the sum of the two velocities is w=v+ 1 u.
E. it will always be true that (ct)2 > x x or (ct)2 − x x = x M x > 0. If we denote by x the travelled way between two events and the elapsed time by t, we will obtain the four-dimensional spacetime interval s from s2 = x M x. 53) The “distance” s between the two events is an invariant interval in the fourdimensional spacetime. 53) is invariant with respect to a Lorentz transformation, s is independent of the chosen inertial frame, it has always the same length. So the theory of relativity does not relativise everything!
Einstein in Matrix Form: Exact Derivation of the Theory of Special and General Relativity without Tensors by Günter Ludyk