Abstract
The earth’s circumference at the equator is about 40,000 km. As there are 24 hours in one day, an hour corresponds to a distance at the equator of about 1666 km and a minute to 28 km.
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Notes
- 1.
The earth’s circumference at the equator is about 40,000 km. As there are 24 hours in one day, an hour corresponds to a distance at the equator of about 1666 km and a minute to 28 km.
- 2.
“Io, Europa, the boy Ganymede, and Callisto greatly pleased lustful Jupiter,” noted Marius in his 1614 publication Mundus Iovialis. There is small doubt that the sex appeal of Marius’s names led to their adoption. Galileo suggested naming the moons Cosimo’s stars in his 1610 publication Sidereus Nuncius, after Cosimo II de’ Medichi, the Grand Duke of Tuscany, who supported Galileo’s work.
- 3.
Galileo’s concerns were not unwarranted. Marius’s student Baldessar Capra published Galileo’s manuscript on operation of the sector (a Galileo invention) under his own name in 1607. When his plagiarism was discovered, Capra was expelled from the University of Padua. Marius, in the meantime, had returned (fled?) to Germany.
- 4.
The long-held geocentric model of Ptolemy was thereby refuted by direct observation of contradictory behavior. Nevertheless, Galileo was ultimately forced to recant his observations in 1633 by the Inquisition and spent his final years under house arrest. Galileo was pardoned in 1992 by Pope John Paul II.
- 5.
From the Greek words \(\acute{\epsilon }\phi \acute{\eta }\mu \epsilon \rho o\zeta\) for “daily” or \(\acute{\epsilon }\phi \acute{\eta }\mu \epsilon \rho \iota \zeta\) for “diary.”
- 6.
Rømer’s presentation was recorded by an anonymous observer.
- 7.
This is the lifetime of free neutrons. Neutrons in the atomic nucleus appear to be stable in the vast majority of cases. A few radioactive isotopes decay via \(\beta\) particle (electron) emission, in which a neutron in the nucleus does decay.
- 8.
Note that \(\mathbf{x}\) and a are not a vectors in our usual sense: the components that make up the vectors need not have the same dimensionality.
- 9.
In these units, the rest mass of a proton is 0.938 GeV/c 2
- 10.
Five standard deviations is now the generally accepted standard of significance in the high energy physics field. Leon Lederman, leader of the E288 collaboration, was undoubtedly chagrined to have the 6.0 GeV/c 2 non-particle deemed the “Oops-Leon” by uncharitable physicists. His subsequent receipt of the 1988 Nobel Prize in Physics (with Melvin Schwartz and Jack Steinberger ) “for the neutrino beam method and discovery of the muon neutrino” undoubtedly softened the blow.
- 11.
Rømer never published his findings in the journal of the Royal Society, in part due to Cassini’s continuing objections to attributing the variations to a finite velocity of light.
- 12.
We shall define these terms in more detail subsequently.
- 13.
The US Strategic Defense Initiative, commonly known as Star Wars, investigated using lasers to shoot down incoming nuclear warheads or destroy missiles in their launch phase. In these studies, the simplifying assumption that the propagating wave does not modify or interact with the medium of propagation has to be abandoned.
- 14.
Note that for these equations to make sense, the coefficients must have dimensions.
- 15.
Setting \(a_{3} = a_{6} = 0\) amounts to making the choice that origins of the coordinate systems coincide at time \(t = t^{{\prime}} = 0\).
- 16.
The parameter ζ here is not related to the radial coordinate in a cylindrical coordinate system.
- 17.
The Dutch physicist Hendrik Antoon Lorentz contributed significantly to our understanding of the mathematical properties of the wave equation. He was awarded the Nobel Prize in Physics (with Pieter Zeeman) in 1902 “in recognition of the extraordinary service they rendered by their researches into the influence of magnetism upon radiation phenomena.”
- 18.
There is a sign choice to be made here. The Lorentz transform also preserves the quantity − s 2. We shall justify our usage soon.
- 19.
It is tempting to think of this as a rotation “around” the time ct axis and to think of the boosts as rotations of a sort around the spatial axes.
- 20.
By positive sense, we use a right-hand rule: if your right thumb points along the axis of rotation, your fingers curl in the positive angular direction.
- 21.
This is not to say that there will never be scientific debate. Cassini, for example, ultimately did not believe in the finite velocity of light, despite his own and Rømer’s data.
- 22.
There are, of course, small perturbations arising from Io’s interactions with the other Jovian moons that affect the period. These are smaller than the precision of Rømer’s experiments.
- 23.
This theory of relativity is special due to the assumption that the transform must be linear. A more general theory of relativity relaxes that restraint.
- 24.
An obvious way to circumvent this limitation is to alter the human life span. If humans could live to be a thousands of years old, then a two-hundred year journey to a distant star might be an appropriate use of one’s time. Such discussions, of course, are more appropriate to classes in molecular biology.
- 25.
The inverse period 1∕T occurs frequently in physics. As a result, it has its own customary name: frequency.
- 26.
As physicists found more and more excitations of matter, they grouped them by mass. Light particles like electrons were called leptons from the Greek word \(\lambda \epsilon \pi \tau \acute{o}\varsigma\) meaning light or thin or delicate. Heavy particles like the proton and neutron were called baryons from the Greek word \(\beta \alpha \rho \acute{\upsilon }\varsigma\) meaning heavy. Mesons were particles of intermediate mass, hence the use of the Greek word \(\mu \acute{\epsilon }\sigma o\nu\) meaning middle.
- 27.
The symbol γ is often used to describe the quantum of the electromagnetic field: the photon. Again, one must be cognizant of the meaning of notation and not attempt to simply memorize equations.
- 28.
This English translation is due to Andrew Motte circa 1729.
- 29.
The SI prefix μ and unit s means 10−6 seconds and is pronounced microseconds. It is a different entity than the muon, which is also represented by the Greek μ.
- 30.
The Nobel Prize in Physics in 2012 was awarded to the Moroccan physicist Serge Haroche and the American physicist David J. Wineland “for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems.” Wineland’s methods were used by the NIST researchers to study relativistic effects at human scale.
- 31.
It is to be expected that non-expert students will often ask the (trivial) question: what is the point? Einstein’s emphasis on the physical interpretation of our mathematics distinguishes the trivial from the more subtle and insightful.
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Cunningham, M.A. (2015). IV On the Nature of Spacetime. In: Neoclassical Physics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-10647-2_4
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