Physics as Spacetime Geometry
... One can call t0 time, but then must necessarily, in connection with this, define space by the manifold of three parameters x0 , y, z in which the laws of physics would then have exactly the same expressions by means of x0 , y, z, t0 as by means of x, y, z, t. Hereafter we would then have in the wor ...
... One can call t0 time, but then must necessarily, in connection with this, define space by the manifold of three parameters x0 , y, z in which the laws of physics would then have exactly the same expressions by means of x0 , y, z, t0 as by means of x, y, z, t. Hereafter we would then have in the wor ...
Vectors and Scalars
... A ship starts its journey at point A and travels for 200 km on a bearing of [50°] to a point B. The ship then changes direction and travels for 100 km on a bearing of [120°] to a point C. Calculate the resultant displacement vector. ...
... A ship starts its journey at point A and travels for 200 km on a bearing of [50°] to a point B. The ship then changes direction and travels for 100 km on a bearing of [120°] to a point C. Calculate the resultant displacement vector. ...
Relativity without tears - Philsci
... Now c is an invariant velocity, as (14) shows. However, in the above derivation nothing as yet indicates that c is the velocity of light. Only when we invoke the Maxwell equations it becomes clear that the velocity of electromagnetic waves implied by these equations must coincide with c if we want t ...
... Now c is an invariant velocity, as (14) shows. However, in the above derivation nothing as yet indicates that c is the velocity of light. Only when we invoke the Maxwell equations it becomes clear that the velocity of electromagnetic waves implied by these equations must coincide with c if we want t ...
PPT - University of Illinois Urbana
... Find the unit normal vector and the differential surface at a point on the surface Find the equation for the direction lines associated with a vector field Identify the polarization of a sinusoidally time-varying vector field Calculate the electric field due to a charge distribution by applying supe ...
... Find the unit normal vector and the differential surface at a point on the surface Find the equation for the direction lines associated with a vector field Identify the polarization of a sinusoidally time-varying vector field Calculate the electric field due to a charge distribution by applying supe ...
Lecture Notes on General Relativity
... originality for its own sake; however, originality sometimes crept in just because I thought I could be more clear than existing treatments. None of the substance of the material in these notes is new; the only reason for reading them is if an individual reader finds the explanations here easier to ...
... originality for its own sake; however, originality sometimes crept in just because I thought I could be more clear than existing treatments. None of the substance of the material in these notes is new; the only reason for reading them is if an individual reader finds the explanations here easier to ...
An introduction to the mechanics of black holes
... Theorem 1 Area law. If (i) Einstein’s equations hold with a matter stress-tensor satisfying the null energy condition, Tµν k µ k ν ≥ 0, for all null k µ , ...
... Theorem 1 Area law. If (i) Einstein’s equations hold with a matter stress-tensor satisfying the null energy condition, Tµν k µ k ν ≥ 0, for all null k µ , ...
7 The Schwarzschild Solution and Black Holes
... note that the result is a static metric. We did not say anything about the source except that it be spherically symmetric. Specifically, we did not demand that the source itself be static; it could be a collapsing star, as long as the collapse were symmetric. Therefore a process such as a supernova ...
... note that the result is a static metric. We did not say anything about the source except that it be spherically symmetric. Specifically, we did not demand that the source itself be static; it could be a collapsing star, as long as the collapse were symmetric. Therefore a process such as a supernova ...
Minkowski space
In mathematical physics, Minkowski space or Minkowski spacetime is a combination of Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity.Minkowski space is closely associated with Einstein's theory of special relativity, and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time will often differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events. Because it treats time differently than the three spacial dimensions, Minkowski space differs from four-dimensional Euclidean space.The isometry group, preserving Euclidean distances of a Euclidean space equipped with the regular inner product is the Euclidean group. The analogous isometry group for Minkowski apace, preserving intervals of spacetime equipped with the associated non-positive definite bilinear form (here called the Minkowski inner product,) is the Poincaré group. The Minkowski inner product is defined as to yield the spacetime interval between two events when given their coordinate difference vector as argument.