Lorentz Scalar - Simple Scalars in Special Relativity - The Length of A Position Vector

The Length of A Position Vector

In special relativity the location of a particle in 4-dimensional spacetime is given by its world line

where is the position in 3-dimensional space of the particle, is the velocity in 3-dimensional space and is the speed of light.

The "length" of the vector is a Lorentz scalar and is given by

where is c times the proper time as measured by a clock in the rest frame of the particle and the metric is given by

eta^{munu} =eta_{munu} = begin{pmatrix}
1 & 0 & 0 & 0\
0 & -1 & 0 & 0\
0 & 0 & -1 & 0\
0 & 0 & 0 & -1
end{pmatrix}.

This is a time-like metric. Often the Minkowski metric is used in which the signs of the ones are reversed.

eta^{munu} =eta_{munu} = begin{pmatrix}
-1 & 0 & 0 & 0\
0 & 1 & 0 & 0\
0 & 0 & 1 & 0\
0 & 0 & 0 & 1
end{pmatrix}.

This is a space-like metric. In the Minkowski metric the space-like interval is defined as

.

We use the Minkowski metric in the rest of this article.

Read more about this topic:  Lorentz Scalar, Simple Scalars in Special Relativity

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