From dba922cd0c8f5ce7252f33268189259706fc9e75 Mon Sep 17 00:00:00 2001 From: bnewbold Date: Sun, 24 Jan 2010 05:23:28 -0500 Subject: partial fixes --- physics/general relativity.page | 24 -------- physics/gravitational waves.page | 117 --------------------------------------- physics/gravitational-waves.page | 117 +++++++++++++++++++++++++++++++++++++++ physics/special relativity.page | 62 --------------------- physics/special-relativity.page | 60 ++++++++++++++++++++ physics/units.page | 14 ++--- 6 files changed, 184 insertions(+), 210 deletions(-) delete mode 100644 physics/general relativity.page delete mode 100644 physics/gravitational waves.page create mode 100644 physics/gravitational-waves.page delete mode 100644 physics/special relativity.page create mode 100644 physics/special-relativity.page diff --git a/physics/general relativity.page b/physics/general relativity.page deleted file mode 100644 index f4a45af..0000000 --- a/physics/general relativity.page +++ /dev/null @@ -1,24 +0,0 @@ ---- -format: rst -categories: physics -toc: no -... - -=========================== -General Relativity -=========================== - -.. warning:: This is a rough work in progress!! Likely to be factual errors, - poor grammar, etc. - -.. note:: Most of this content is based on a 2002 Caltech course taught by - Kip Thorn [PH237]_ - -*See also `math/tensors `__* - -(no content) - -References ----------------- - -.. [PH237] `Gravitational Waves`:title: (aka ph237), a course taught by Kip Thorne at Caltech in 2002. See http://elmer.tapir.caltech.edu/ph237/ for notes and lecture videos. diff --git a/physics/gravitational waves.page b/physics/gravitational waves.page deleted file mode 100644 index 66f6c04..0000000 --- a/physics/gravitational waves.page +++ /dev/null @@ -1,117 +0,0 @@ ---- -format: rst -categories: physics -toc: no -... - -======================= -Gravitational Waves -======================= - -.. warning:: This is a rough work in progress!! Likely to be factual errors, poor grammar, etc. - -.. note:: Most of this content is based on a 2002 Caltech course taught by - Kip Thorn [PH237]_ - -Raw Info ------------------ -Rank 4 Riemann tensors, will cover different gauge. -Waves are double integrals of curvature tensor... - - - -Gravitons as Quantum Particles ---------------------------------- -Invariance angles: (Spin of quantum particle) = :latex:`$2 pi$` / (invariance angle) - -Graviton has :latex:`$\pi$` invariance angle, so it is spin 2; photons have unique :latex:`$\arrow{E}$` vector, so invariance angle is :latex:`$2\pi$`, spin 1 - -Also describes spin by the group of Lorentz transformations which effect propagation. - -Two polarizations: cross and plus, corresponding to spin of particles aligning with or against propagation? (Ref: Eugene Vickner? reviews of modern physics) - -Waves' multipole order :latex:`$\geq$` spin of quantum = 2 for graviton ((??)) - -Waves don't propagate like E, because mass monopoles don't oscillate like charges. - -:latex:`$ h \approx \frac{G}{c^2} \frac{M_0}{r} + \frac{G}{c^3} \frac{M'_1}{r} + \frac{G}{c^4} \frac{M''_2}{r} + \frac{G}{c^4} \frac{S'_1}{r} + \frac{G}{c^5} \frac{S''_1}{r}$` -First term: mass can't oscillate -Second term: momentum can't oscillate -Third term: mass quadrupole moment dominates -Fourth term: angular momentum can't oscillate -Fifth term: current quadrupole - -Energy ----------------- - -Quick calculation: for a source with mass M, size L, period P, the quadrupole -moment :m:`$M_2 \approx M L^2$`, :m:`$h \approx 1/c^2` (Newtonian potential -energy) ???? - -h on the order of :m:`$10^{-22}$` - -Propagation ------------------ - -When wavelength much less than curvature of universe (background), then gravitational waves propagate like light waves: undergo red shifts, gravitational lensing, inflationary red shift, etc. - -Sources -------------- - -Inspirals of bodies into super-massive black holes - Eg, white dwarfs, neutron stars, small black holes. - Super-massive black holes are expected near the centers of galaxies. - Low frequencies (LISA); waveforms could hold data about spacetime curvature - local to the black hole. - Waveforms could be very difficult to predict. - -Binary black hole mergers - Broadband signals depending on masses. - -Neutron Star/Black hole mergers - Stellar mass objects existing in the main bodies of galaxies. - Higher frequencies (LIGO and AdvLIGO). - -Neutron Star/Neutron Star mergers - Have actual examples in our galaxy of these events; but final inspiral rate - is so low that we have must listen in other galaxies. Merger waves will - probably be lost in higher frequency noise, so can't probe local - gravitational curvature. - May observe "tails" of waves: scattering off of high curvature around the - binary. - -Pulsars (spinning neutron stars) - Known to exist in our galaxy. - -Spectrum ----------------- - -High Frequency: Above 1 Hz, LIGO (10 Hz to 1kHz), resonant bars - Small black holes (2 to 1k suns), neutron stars, supernovas - -Low frequency: 1Hz and lower, LISA (10^-4 Hz to 0.1 Hz), Doppler tracking of spacecraft - Massive black holes (300 to 30 million suns), binary stars - -Very Low Frequency: 10^-8 Hz, Pulsar timing (our clocks shifted by gwaves, average of distance pulsars are not over long periods) - -Extreme Low Frequency: 10^-16 Hz, Cosmic Microwave Background anisotropy - -Detectors ------------------ - -:m:`$\Delta L = h L ~ \leq 4 \times 10^{-16} \text{cm}$` - -LIGO (10 Hz to 1kHz) - Also GEO, VIRGO, TAMA (?), AIGO - -LISA (10e-4 Hz to 0.1 Hz) - -Resonant Bars -~~~~~~~~~~~~~~~ -First by Webber. -Currently in Louisiana State University (Allegro), University of West Australia (Niobe), CERN (Explorer), University of Padova (Auriga), and University of Rome (Nautilus) - -References ----------------- - -.. [PH237] `Gravitational Waves`:title: (aka ph237), a course taught by Kip Thorne at Caltech in 2002. See http://elmer.tapir.caltech.edu/ph237/ for notes and lecture videos. diff --git a/physics/gravitational-waves.page b/physics/gravitational-waves.page new file mode 100644 index 0000000..c853e2b --- /dev/null +++ b/physics/gravitational-waves.page @@ -0,0 +1,117 @@ +--- +format: rst +categories: physics +toc: no +... + +======================= +Gravitational Waves +======================= + +.. warning:: This is a rough work in progress!! Likely to be factual errors, poor grammar, etc. + +.. note:: Most of this content is based on a 2002 Caltech course taught by + Kip Thorn [PH237]_ + +Raw Info +----------------- +Rank 4 Riemann tensors, will cover different gauge. +Waves are double integrals of curvature tensor... + + + +Gravitons as Quantum Particles +--------------------------------- +Invariance angles: (Spin of quantum particle) = $2 pi$ / (invariance angle) + +Graviton has $\pi$ invariance angle, so it is spin 2; photons have unique $\arrow{E}$ vector, so invariance angle is $2\pi$, spin 1 + +Also describes spin by the group of Lorentz transformations which effect propagation. + +Two polarizations: cross and plus, corresponding to spin of particles aligning with or against propagation? (Ref: Eugene Vickner? reviews of modern physics) + +Waves' multipole order $\geq$ spin of quantum = 2 for graviton ((??)) + +Waves don't propagate like E, because mass monopoles don't oscillate like charges. + +$ h \approx \frac{G}{c^2} \frac{M_0}{r} + \frac{G}{c^3} \frac{M'_1}{r} + \frac{G}{c^4} \frac{M''_2}{r} + \frac{G}{c^4} \frac{S'_1}{r} + \frac{G}{c^5} \frac{S''_1}{r}$ + +First term: mass can't oscillate, +Second term: momentum can't oscillate, +Third term: mass quadrupole moment dominates, +Fourth term: angular momentum can't oscillate, +Fifth term: current quadrupole + +Energy +---------------- + +Quick calculation: for a source with mass M, size L, period P, the quadrupole +moment $M_2 \approx M L^2$, $h \approx 1/c^2 (Newtonian potential energy) ???? + +h on the order of $10^{-22}$ + +Propagation +----------------- + +When wavelength much less than curvature of universe (background), then gravitational waves propagate like light waves: undergo red shifts, gravitational lensing, inflationary red shift, etc. + +Sources +------------- + +Inspirals of bodies into super-massive black holes + Eg, white dwarfs, neutron stars, small black holes. + Super-massive black holes are expected near the centers of galaxies. + Low frequencies (LISA); waveforms could hold data about spacetime curvature + local to the black hole. + Waveforms could be very difficult to predict. + +Binary black hole mergers + Broadband signals depending on masses. + +Neutron Star/Black hole mergers + Stellar mass objects existing in the main bodies of galaxies. + Higher frequencies (LIGO and AdvLIGO). + +Neutron Star/Neutron Star mergers + Have actual examples in our galaxy of these events; but final inspiral rate + is so low that we have must listen in other galaxies. Merger waves will + probably be lost in higher frequency noise, so can't probe local + gravitational curvature. + May observe "tails" of waves: scattering off of high curvature around the + binary. + +Pulsars (spinning neutron stars) + Known to exist in our galaxy. + +Spectrum +---------------- + +High Frequency: Above 1 Hz, LIGO (10 Hz to 1kHz), resonant bars + Small black holes (2 to 1k suns), neutron stars, supernovas + +Low frequency: 1Hz and lower, LISA (10^-4 Hz to 0.1 Hz), Doppler tracking of spacecraft + Massive black holes (300 to 30 million suns), binary stars + +Very Low Frequency: 10^-8 Hz, Pulsar timing (our clocks shifted by gwaves, average of distance pulsars are not over long periods) + +Extreme Low Frequency: 10^-16 Hz, Cosmic Microwave Background anisotropy + +Detectors +----------------- + +$\Delta L = h L ~ \leq 4 \times 10^{-16} \text{cm}$ + +LIGO (10 Hz to 1kHz) + Also GEO, VIRGO, TAMA (?), AIGO + +LISA (10e-4 Hz to 0.1 Hz) + +Resonant Bars +~~~~~~~~~~~~~~~ +First by Webber. +Currently in Louisiana State University (Allegro), University of West Australia (Niobe), CERN (Explorer), University of Padova (Auriga), and University of Rome (Nautilus) + +References +---------------- + +[PH237]: **Gravitational Waves** (aka ph237), a course taught by Kip Thorne at Caltech in 2002. See http://elmer.tapir.caltech.edu/ph237/ for notes and lecture videos. diff --git a/physics/special relativity.page b/physics/special relativity.page deleted file mode 100644 index 2f5b9af..0000000 --- a/physics/special relativity.page +++ /dev/null @@ -1,62 +0,0 @@ ---- -format: rst -categories: physics -toc: no -... - -=========================== -Special Relativity -=========================== - -.. warning:: This is a rough work in progress!! Likely to be factual errors, - poor grammar, etc. - -.. note:: Most of this content is based on a 2002 Caltech course taught by - Kip Thorn [PH237]_ - -*See also `physics/general relativity`__* - -As opposed to general relativity, special relativity takes place in a *flat* -Minkowski space time: a 4-space with three spatial dimensions and one time -dimension. - -+----------------+--------------------+ -| Index notation | Variable | Type | -+----------------+--------------------+ -| `$x^0$`:m: | `$t$`:m: | Time | -| `$x^1$`:m: | `$x$`:m: | Spatial | -| `$x^2$`:m: | `$y$`:m: | Spatial | -| `$x^3$`:m: | `$z$`:m: | Spatial | -+----------------+--------------------+ - -Separations -------------- - -The separation `$(\Delta s)^2$`:m: between two events in space time, in a given -Lorentzian/inertial frame, is defined -as: - -:m:`$$ (\Delta s)^2 \equiv -(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 $$` - -or - -:m:`$$ (\Delta s)^2 \equiv -(\Delta x^0)^2 + \sum_{i,j} \delta_{ij} \Delta x^i \Delta x^j$$` - -where :m:`$\delta_{ij}$` is the Kronecker delta (unity or 1 when -:m:`$i=j$`; zero otherwise), and the indices i and j are over the spatial -dimensions 1,2,3 (corresponding to x,y,z). It can be shown that this separation -is Lorentz-invariant; the scalar value of separation between two events does -not depend on the inertial frame chosen. - -Note the negative sign in front of the time dimension. The are three types of -separations: **space-like** when :m:`$(\Delta s)^2 > 0$`, **null-** or -**light-like** when :m:`$(\Delta s)^2 = 0$`, and **time-like** when -:m:`$(\Delta s)^2 < 0$`. When dealing with time-like separations, ignore the -implication of an imaginary number. The difference in time :m:`$\Delta \Tau$` -is always real: :m:`($\Delta \Tau)^2= -(\Delta s)^2$`. - - -References ----------------- - -.. [PH237] `Gravitational Waves`:title: (aka ph237), a course taught by Kip Thorne at Caltech in 2002. See http://elmer.tapir.caltech.edu/ph237/ for notes and lecture videos. diff --git a/physics/special-relativity.page b/physics/special-relativity.page new file mode 100644 index 0000000..bb5f564 --- /dev/null +++ b/physics/special-relativity.page @@ -0,0 +1,60 @@ +--- +format: rst +categories: physics +toc: no +... + +=========================== +Special Relativity +=========================== + +.. warning:: This is a rough work in progress!! Likely to be factual errors, + poor grammar, etc. + +.. note:: Most of this content is based on a 2002 Caltech course taught by + Kip Thorn [PH237]_ + +As opposed to general relativity, special relativity takes place in a *flat* +Minkowski space time: a 4-space with three spatial dimensions and one time +dimension. + +-------------- -------- --------- +Index notation Variable Type +-------------- -------- --------- + $x^0$ $t$ Time + $x^1$ $x$ Spatial + $x^2$ $y$ Spatial + $x^3$ $z$ Spatial +-------------- -------- --------- + +Separations +------------- + +The separation $(\Delta s)^2$ between two events in space time, in a given +Lorentzian/inertial frame, is defined +as: + +$$ (\Delta s)^2 \equiv -(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 $$ + +or + +$$ (\Delta s)^2 \equiv -(\Delta x^0)^2 + \sum_{i,j} \delta_{ij} \Delta x^i \Delta x^j$$ + +where $\delta_{ij}$ is the Kronecker delta (unity or 1 when +$i=j$; zero otherwise), and the indices i and j are over the spatial +dimensions 1,2,3 (corresponding to x,y,z). It can be shown that this separation +is Lorentz-invariant; the scalar value of separation between two events does +not depend on the inertial frame chosen. + +Note the negative sign in front of the time dimension. The are three types of +separations: **space-like** when $(\Delta s)^2 > 0$, **null-** or +**light-like** when $(\Delta s)^2 = 0$, and **time-like** when +$(\Delta s)^2 < 0$. When dealing with time-like separations, ignore the +implication of an imaginary number. The difference in time $\Delta \Tau$ +is always real: ($\Delta \Tau)^2= -(\Delta s)^2$. + + +References +---------------- + +[PH237]: **Gravitational Waves** (aka ph237), a course taught by Kip Thorne at Caltech in 2002. See http://elmer.tapir.caltech.edu/ph237/ for notes and lecture videos. diff --git a/physics/units.page b/physics/units.page index 385136c..bfc78bc 100644 --- a/physics/units.page +++ b/physics/units.page @@ -3,11 +3,11 @@ format: rst categories: physics toc: no ... + ====================== Units ====================== -.. contents:: SI Units -------------------- @@ -26,7 +26,7 @@ Natural Units Natural units are a system of units which replace (or re-scale) the usual mass, length, and time bases with quantities which have "natural" (physical) constants associated with them. The two constants usually chosen are the speed -of light (c) and Plank's constant (:m:`$\hbar$`); the gravitational constant +of light (c) and Plank's constant ($\hbar$); the gravitational constant (G) is a possibility for the third constant/unit, but energy (in electron-volts: eV) is often used instead because it gives more useful relations and because there is no accepted theory of quantum gravity to unite @@ -36,18 +36,18 @@ Working with natural units simplifies physical relations and equations because many conversion factors drop out. Given the relations between cgs units (gm, cm, sec) and natural units (c, -:m:`$\hbar$` , eV), we can find the natural units of an arbitrary quantity -:m:`$[Q]=[gm]^{a}[cm]^{b}[sec]^{c}=[c]^{\alpha}[\hbar]^{\beta}[eV]^{\gamma}$`: +$\hbar$ , eV), we can find the natural units of an arbitrary quantity +$[Q]=[gm]^{a}[cm]^{b}[sec]^{c}=[c]^{\alpha}[\hbar]^{\beta}[eV]^{\gamma}$: -:m:`$$(\alpha,\beta,\gamma)=\left(\begin{array}{ccc} -2 & 1 & 0\\ 0 & 1 & 1\\ 1 & -1 & -1\end{array}\right)\left(\begin{array}{c} a\\ b\\ c\end{array}\right)=(-2a+b,b+c,a-b-c)$$` +$$(\alpha,\beta,\gamma)=\left(\begin{array}{ccc} -2 & 1 & 0\\ 0 & 1 & 1\\ 1 & -1 & -1\end{array}\right)\left(\begin{array}{c} a\\ b\\ c\end{array}\right)=(-2a+b,b+c,a-b-c)$$ or in reverse: -:m:`$$(a,b,c)=\left(\begin{array}{ccc} 0 & 1 & 1\\ 1 & 2 & 2\\ -1 & -1 & -2\end{array}\right)\left(\begin{array}{c} \alpha\\ \beta\\ \gamma\end{array}\right)=(\beta+\gamma,\alpha+2\beta+\gamma,-\alpha-\beta-2\gamma)$$` +$$(a,b,c)=\left(\begin{array}{ccc} 0 & 1 & 1\\ 1 & 2 & 2\\ -1 & -1 & -2\end{array}\right)\left(\begin{array}{c} \alpha\\ \beta\\ \gamma\end{array}\right)=(\beta+\gamma,\alpha+2\beta+\gamma,-\alpha-\beta-2\gamma)$$ Plank Units ---------------- -Plank units (defined by Plank soon after defining his constant :m:`$\hbar$`) are a version of _`Natural Units` using the gravitational constant G as the the +Plank units (defined by Plank soon after defining his constant $\hbar$) are a version of _`Natural Units` using the gravitational constant G as the the third unit (instead of the common measure of energy). When converted back into mass-length-time units we get three quantities which define the "Plank Scale", which may provide estimation of the domain where quantum gravity effects become -- cgit v1.2.3