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For ranks greater than two, the symmetric or antisymmetric index pairs must be explicitly identified. Antisymmetric tensors of rank 2 play important roles in relativity theory. The set of all such tensors - often called bivectors — forms a vector space of dimension 6, sometimes called bivector space.
This gives ∂y 1 / ∂x = −sin x / r and ∂y 2 / ∂x = cos x / r In this case the metric is a scalar and is given by g = cos 2 x / r + sin 2 x / r = 1. The interval is then ds 2 = g dx 2 = dx 2. The interval is just equal to the arc length as expected.
Fig 2-1 In the theory of relativity each observer assigns the event at A to a different time and location. Fig 2-2 Minkowski diagram for various speeds of the primed frame, which is moving relative to the unprimed frame. The dashed lines represent the light cone of a flash of light at the origin.
Relativity links mass with energy, and energy with momentum. The equivalence between mass and energy, as expressed by the formula E = mc 2, is the most famous consequence of special relativity. In relativity, mass and energy are two different ways of describing one physical quantity.
If one is only interested in the weak field limit of the theory, the dynamics of matter can be computed using special relativity methods and/or Newtonian laws of gravity and the resulting stress–energy tensor can then be plugged into the Einstein field equations. But if one requires an exact solution or a solution describing strong fields ...
Hermann Minkowski (1864–1909) found that the theory of special relativity could be best understood as a four-dimensional space, since known as the Minkowski spacetime.. In physics, Minkowski space (or Minkowski spacetime) (/ m ɪ ŋ ˈ k ɔː f s k i,-ˈ k ɒ f-/ [1]) is the main mathematical description of spacetime in the absence of gravitation.
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics.
(Note that this does not apply in non-inertial frames, indeed between accelerating frames the speed of light cannot be constant. [1] Although it can be applied in non-inertial frames if an observer is confined to making local measurements. [2]) Einstein developed the theory of special relativity based on these two postulates.