Abstract
The aim of physics is to describe the laws underlying physical phenomena.
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Notes
- 1.
The reference frame associated with an observer is defined by a coordinate system, which we shall choose to be a system of rectangular Cartesian coordinates (x, y, z) with origin O, with respect to which the observer is at rest. The frame also consists of all the instruments the observer needs for measuring the fundamental quantities: a ruler for lengths, a clock for time intervals and scales for masses.
- 2.
Here we are referring to fundamental forces of the nature, like the gravitational force, not to phenomenological forces like elastic forces, friction etc.
- 3.
In general we call covariant an equation which takes the same form in different frames; if not just the form, but also the numerical values of the various terms are the same, we then say that the equation is invariant.
- 4.
Another reason for ruling out the emitting source as the privileged frame where the Maxwell equations hold is the fact that, according to the laws of electromagnetism, an electric charge in an electric field \(\mathbf{E}\) acquires an acceleration \(\mathbf{a}\) which must vanish when \(\mathbf{E}\rightarrow 0\); when the frame is accelerated, as it is generally the case for a moving source, \(\mathbf{a}\) would preserve a non vanishing component equal to acceleration of the reference frame even in the limit of vanishing electric field.
- 5.
Here special refers to the fact that it is restricted to inertial frames.
- 6.
To have an idea of this approximation, consider a very high velocity like, for instance, that of the earth around the sun, which is about \(V\approx 3\times 10^4\)Â m/s. In this case we have \(V^2/c^2\approx 10^{-8}\).
- 7.
The reader should not mistake the upper labels of the space-time coordinates \(x^0,x^1,x^2,x^3\) as powers of a quantity x! The mathematical difference between quantities labeled by upper and lower indices will be extensively discussed in the following chapters.
- 8.
Note that the time dilation is a relative effect, that is if we have a clock at rest in S, from Eq. (1.61) it follows that \(\varDelta t' = \gamma (V)\,\varDelta t\), that is time in \(S'\) is dilated with respect to S. The same observation applies to the length contraction to be discussed in the following.
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D’Auria, R., Trigiante, M. (2016). Special Relativity. In: From Special Relativity to Feynman Diagrams. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-22014-7_1
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DOI: https://doi.org/10.1007/978-3-319-22014-7_1
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