I seem to always forget about the properties concerning inequalities and absolute values
##{\displaystyle |a|\leq b\iff -b\leq a\leq b}##
##{\displaystyle |a|\geq b\iff a\leq -b\ } ## or ## {\displaystyle b\leq a}##
I always fail to remember with which one is which... :cry:
Yeah for some reason I assumed this happened near you but then I read again that you heard the thunder later on, so if you were near the lighting this would have happened almost simultaneously :P
##-d \tau^2=ds^2=-(1-\frac{2M}{r})dt^2+
r^2 d\phi ^2## →##d \tau^2 = [(1-\frac{2M}{r})(\frac{dt}{d \phi})^2-r^2]dφ^2## and for circular orbits like the one assumed around the static black hole ##Ω^2=(\frac{d φ}{dt})^2=\frac{M}{r^3}## so you have
##Δτ=\int_{0}^{2π} \sqrt{\frac{r}{M}-3}...
lets say we have three observers, a) one in moving around a static (Schwarzschild) black hole at a radius let's say r=10M b) one that is at infinity and gets light signals from the first observer and c) an observer that is somehow standing still at radius r=10M and also sees the light signals...
Like Ibix said a coordinate singularity is just a bad choice we made.
But for the Schwarzschild metric which is a solution to the Einstein field equations and describes the vacuum outside of spherical mass distributions, the metric element contains two singularities ##r=0## which is an...
a singularity is the point where all physics laws break down.
For example in the Schwarzschild metric, there are 2 anomalies in the radial coordinate ##r=0## as a pure singularity and another one ##r=2M## but that's just a coordinate singularity.
The Schwarzschild metric is given by
##ds^2 = -...
Maybe one should ask where does the mass of a black hole lie. Of course black holes are made of pure spacetime and nothing else; but where exactly does this mass lie, since the matter from the progenitor star is gone and all is left is the singularity and other than that the properties of the...
You mean Thoma's book, I guess. Now for Physics textbooks, what kind of books are interested in? You can have a look at Young & Freedman University Physics but there are numerous books, it really depends on what you are looking for.
That's okay then.
The only book I recommended was the one about calculus, so good luck with this one! For any further questions do not hesitate to ask again.
This book has Mechanics, Waves and Thermodynamics so if you can actually do all these then you can do a first year lectures at a physics department of a university.
About Mathematics now, make sure you have a solid (and when I say solid I mean it) background in calculus, try the book ""Thomas'...