Metric and transformations of coordinates
Having recognised the four-dimensional nature of spacetime, we are driven to employ the Minkowski metric, η, given in components (valid in any inertial reference frame In physics, an inertial frame of reference is a reference frame, tied to the state of motion of an observer, with the property that each physical law portrays itself in the same form in every inertial frame. The contrasting case is the set of non-inertial frames, in which the laws of physics change from frame to frame, and the usual forces) as:
which is equal to its reciprocal, ηαβ, in those frames.
Then we recognize that coordinate transformations between inertial reference frames are given by the Lorentz transformation In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in uniform motion with respect to the other. In classical physics (Galilean relativity), the only conversion believed necessary was x' = x − vt, describing how the origin of one observer's coordinate system tensor A tensor is an object which includes and extends the notion of scalar, vector, and matrix. The term is slightly ambiguous, and this often leads to misunderstandings; roughly speaking, there are different default meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is first of Λ. For the special case of motion along the x-axis, we have:
which is simply the matrix of a boost (like a rotation) between the x and ct coordinates. Where μ' indicates the row and ν indicates the column. Also, β and γ are defined as:
More generally, a transformation from one inertial frame (ignoring translations for simplicity) to another must satisfy:
where there is an implied summation of and from 0 to 3 on the right-hand side in accordance with the Einstein summation convention In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulas. It was introduced by Albert Einstein in 1916. The Poincaré group In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime. It is a 10-dimensional noncompact Lie group. The abelian group of translations is a normal subgroup while the Lorentz group is a subgroup, the stabilizer of a point. That is, the full Poincaré group is the affine group is the most general group of transformations which preserves the Minkowski metric and this is the physical symmetry underlying special relativity.
All proper physical quantities are given by tensors. So to transform from one frame to another, we use the well-known tensor transformation law A tensor is an object which includes and extends the notion of scalar, vector, and matrix. The term is slightly ambiguous, and this often leads to misunderstandings; roughly speaking, there are different default meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is first of
Where is the reciprocal matrix of .
To see how this is useful, we transform the position of an event from an unprimed coordinate system S to a primed system S', we calculate
which is the Lorentz transformation given above. All tensors transform by the same rule.
The squared length of the differential of the position four-vector constructed using
is an invariant. Being invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. Notice that when the line element A line element in mathematics can most generally be thought of as the square of the change in a position vector in an affine space equated to the square of the change of the arc length. An easy way of visualizing this relationship is by parametrizing the given curve by Frenet's formulas. As such, a line element is then naturally a function of the is negative that is the differential of proper time In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a shorter proper time between two events than a non-accelerated clock between the same events. The twins, while when is positive, is differential of the proper distance In relativistic physics, proper length is an invariant quantity which is the rod distance between spacelike-separated events in a frame of reference in which the events are simultaneous.
The primary value of expressing the equations of physics in a tensor form is that they are then manifestly invariant under the Poincaré group, so that we do not have to do a special and tedious calculation to check that fact. Also in constructing such equations we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation.
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