This Insight develops equations of state that are useful in calculations about cosmology and about the insides of stars. The first calculation is for a photon gas and the second is for a ‘relativistic’ gas of particles with mass. Photon Gas Exercise 22 on p108 of Bernard Schutz’s ‘The first course in General Relativity (Second…
This Insight is part of my attempt to develop a formal definition of ‘large-scale isotropy’, a concept that is fundamental to most cosmology, but that is nowhere that I have seen properly defined. The definitions of isotropy are as precise as one could wish, but the ‘large-scale’ bit is in every case I have seen…
Local energy conservation in general relativity (GR), expressed as ##{T^{0b}}_{;b}=0##, combines with the Friedmann-Lemaître-Robertson-Walker (FLRW) metric to yield the relation ##\frac{d}{dt}(\rho R^3)=-p\frac{d}{dt}(R^3)##, where ##\rho## is energy density, ##p## is pressure, and ##R## is the scale factor. This equation, together with the Einstein field equation, is used to derive the Friedmann equations governing the dynamics of…
Differentiation is usually taught quite well. Perhaps that’s because it is the first introduction to calculus, which is considered a big step in a student’s mathematical education, so teachers mostly take a lot of care with that step. Unfortunately, the same cannot be said for partial differentiation. It is often glossed over in a rush…
Introduction Purpose and scope In late high school physics courses and first-year university courses, the phenomenon of acoustical ‘beats’ (digital audio) is often covered. This is an interesting phenomenon, but it is usually only skated over, leaving the students with not much more than a feeling that the beats somehow arise from the two waves…