From f61026119df4700f69eb73e95620bc5928ca0fcb Mon Sep 17 00:00:00 2001 From: User Date: Tue, 13 Oct 2009 02:52:09 +0000 Subject: Grand rename for gitit transfer --- physics/LIGO | 48 ----------------- physics/LIGO.page | 48 +++++++++++++++++ physics/general relativity | 18 ------- physics/general relativity.page | 18 +++++++ physics/gravitational waves | 111 --------------------------------------- physics/gravitational waves.page | 111 +++++++++++++++++++++++++++++++++++++++ physics/quantum/fermigas | 51 ------------------ physics/quantum/fermigas.page | 51 ++++++++++++++++++ physics/special relativity | 56 -------------------- physics/special relativity.page | 56 ++++++++++++++++++++ physics/units | 53 ------------------- physics/units.page | 53 +++++++++++++++++++ 12 files changed, 337 insertions(+), 337 deletions(-) delete mode 100644 physics/LIGO create mode 100644 physics/LIGO.page delete mode 100644 physics/general relativity create mode 100644 physics/general relativity.page delete mode 100644 physics/gravitational waves create mode 100644 physics/gravitational waves.page delete mode 100644 physics/quantum/fermigas create mode 100644 physics/quantum/fermigas.page delete mode 100644 physics/special relativity create mode 100644 physics/special relativity.page delete mode 100644 physics/units create mode 100644 physics/units.page (limited to 'physics') diff --git a/physics/LIGO b/physics/LIGO deleted file mode 100644 index ce682b4..0000000 --- a/physics/LIGO +++ /dev/null @@ -1,48 +0,0 @@ -======================================================================= -LIGO: Laser Interferometer Gravitational Observatory -======================================================================= - -.. 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]_ - -Noise Sources -~~~~~~~~~~~~~~~~~~~~~~ - -For initial LIGO, seismic noise dominates below about 60Hz, suspension thermal -noise between 60 and 180Hz, and radiation pressure shot noise above 180Hz. - -Sensitivity -~~~~~~~~~~~~~~~~~~~~~ - -Advanced LIGO will use 40kg sapphire test masses with sensitivity of about 10e-19 meters: 1/10000 of an atomic nucleus, 10e-13 of a wavelength, and half of the entire mirror's wave function. - -LISA -~~~~~~~~~~~~~ - -5e6 km separations between three spacecraft, 1 (astronomical unit, ~1.5e8 from the sun. 1 watt lasers. -The heterodyne detection is of the beat frequencies at each spacecraft of the -two incoming beams. Doppler shifts of spacecraft must be taken into account, -due not only to sun radiation pressure etc, but varying gravitational fields -from planetary orbits. - -The test masses inside LISA should be free falling and have relative -separations stable to 10e-9 cm (10e-5 wavelength of light). - -LISA's sensitivity is in the milihertz regime. - -.. note: (insert LISA noise curve?) - -Data Analysis -~~~~~~~~~~~~~~~~~~~~ - -Using matched filtering (eg, take cross correlation between two waveforms, -integrating their product), frequency sensitivity will be around the inverse -of the number of cycles of waveform (for LIGO, around 20,000 cycles, for LISA -around 200,000 cycles). - -This technique requires known, theoretically derived waveforms (within -phase/amplitude). There are other methods when we don't have good guesses -about the waveform we are looking for... - diff --git a/physics/LIGO.page b/physics/LIGO.page new file mode 100644 index 0000000..ce682b4 --- /dev/null +++ b/physics/LIGO.page @@ -0,0 +1,48 @@ +======================================================================= +LIGO: Laser Interferometer Gravitational Observatory +======================================================================= + +.. 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]_ + +Noise Sources +~~~~~~~~~~~~~~~~~~~~~~ + +For initial LIGO, seismic noise dominates below about 60Hz, suspension thermal +noise between 60 and 180Hz, and radiation pressure shot noise above 180Hz. + +Sensitivity +~~~~~~~~~~~~~~~~~~~~~ + +Advanced LIGO will use 40kg sapphire test masses with sensitivity of about 10e-19 meters: 1/10000 of an atomic nucleus, 10e-13 of a wavelength, and half of the entire mirror's wave function. + +LISA +~~~~~~~~~~~~~ + +5e6 km separations between three spacecraft, 1 (astronomical unit, ~1.5e8 from the sun. 1 watt lasers. +The heterodyne detection is of the beat frequencies at each spacecraft of the +two incoming beams. Doppler shifts of spacecraft must be taken into account, +due not only to sun radiation pressure etc, but varying gravitational fields +from planetary orbits. + +The test masses inside LISA should be free falling and have relative +separations stable to 10e-9 cm (10e-5 wavelength of light). + +LISA's sensitivity is in the milihertz regime. + +.. note: (insert LISA noise curve?) + +Data Analysis +~~~~~~~~~~~~~~~~~~~~ + +Using matched filtering (eg, take cross correlation between two waveforms, +integrating their product), frequency sensitivity will be around the inverse +of the number of cycles of waveform (for LIGO, around 20,000 cycles, for LISA +around 200,000 cycles). + +This technique requires known, theoretically derived waveforms (within +phase/amplitude). There are other methods when we don't have good guesses +about the waveform we are looking for... + diff --git a/physics/general relativity b/physics/general relativity deleted file mode 100644 index 7fc29eb..0000000 --- a/physics/general relativity +++ /dev/null @@ -1,18 +0,0 @@ -=========================== -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/general relativity.page b/physics/general relativity.page new file mode 100644 index 0000000..7fc29eb --- /dev/null +++ b/physics/general relativity.page @@ -0,0 +1,18 @@ +=========================== +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 b/physics/gravitational waves deleted file mode 100644 index 5aa1744..0000000 --- a/physics/gravitational waves +++ /dev/null @@ -1,111 +0,0 @@ -======================= -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..5aa1744 --- /dev/null +++ b/physics/gravitational waves.page @@ -0,0 +1,111 @@ +======================= +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/quantum/fermigas b/physics/quantum/fermigas deleted file mode 100644 index 0114b43..0000000 --- a/physics/quantum/fermigas +++ /dev/null @@ -1,51 +0,0 @@ -=============== -Fermi Gas -=============== - -Derivation of the Fermi Energy ---------------------------------- -Consider a crystal lattice with an electron gas as a 3 dimensional infinite -square well with dimensions :m:`$l_{x}, l_{y}, l_z$`. The wavefunctions of -individual fermions (pretending they are non-interacting) can be seperated -as :m:`$\psi(x,y)=\psi_{x}(x)\psi_{y}(y)\psi_{z}(z)$`. The solutions will be -the usual ones to the Schrodinger equation: - -:m:`$$\frac{-\hbar^2}{2m}\frac{d^2 \psi_x}{dx}=E_x \psi_x$$` - -with the usual wave numbers :m:`$k_x=\frac{\sqrt{2mE_x}}{\hbar}$`, and quantum -numbers satisfying the boundry conditions :m:`$k_x l_x = n_x \pi$`. The full -wavefunction for each particle will be: - -:m:`$$\psi_{n_{x}n_{y}n_{z}}(x,y,z)=\sqrt{\frac{4}{l_{x}l_{y}}}\sin\left(\frac{n_{x}\pi}{l_{x}}x\right)\sin\left(\frac{n_{y}\pi}{l_{y}}y\right)\sin\left(\frac{n_{z}\pi}{l_{z}}z\right)$$` - -and the associated energies (with :m:`$E = E_x + E_y + E_z$`): - -:m:`$$E_{n_{x}n_{y}n_z}=\frac{\hbar^{2}\pi^{2}}{2m}\left(\frac{n_{x}^{2}}{l_{x}^{2}}+\frac{n_{y}^{2}}{l_{y}^{2}}+\frac{n_{z}^{2}}{l_{z}^{2}}\right)=\frac{\hbar^2|\vec{k}|^2}{2m}$$` - -where :m:`$|\vec{k}|^2$` is the magnitude of the particle's k-vector in k-space. -This k-space can be imagined as a grid of blocks, each representing a possible -particle state (with a double degeneracy for spin). Positions on this grid have -coordinates :m:`$(k_{x},k_{y},k_z)$` corresponding to the positive integer -quantum numbers. These blocks will be filled -from the lowest energy upwards: for large numbers of occupying particles, -the filling pattern can be approximated as an expanding spherical shell with -radius :m:`$|\vec{k_F}|^2$`. - -Note that we're "over counting" the number of occupied states because the -"sides" of the quarter sphere in k-space (where one of the associated quantum -numbers is zero) do not represent valid states. These surfaces can be ignored -for very large N because the surface area to volume ratio is so low, but the -correction can be important. There will then be a second correction due to -removing the states along the individual axes twice (once for each -side-surface), u.s.w. - -The surface of this shell is called the Fermi surface and represents the most -excited states in the gas. The radius can be derived by calculating the total -volume enclosed: each block has volume :m:`$\frac{\pi^3}{l_x l_y -l_z}=\frac{\pi^3}{V}$` and there are N/2 blocks occupied by N fermions, so: - -:m:`$$\frac{1}{8}(\frac{4\pi}{3} |k_{F}|^{3})&=&\frac{Nq}{2}(\frac{\pi^{3}}{V})\\|k_{F}|&=&\sqrt{\frac{3Nq\pi^2}{V}}^3=\sqrt{3\pi^2\rho}^3$$` - -:m:`$\rho$` is the "free fermion density". The corresponding energy is: - -:m:`$$E_{F}=\frac{\hbar^{2}}{2m}|k_{F}|^{2}=\frac{\hbar^{2}}{2m}(3\rho \pi)^{2/3}$$` diff --git a/physics/quantum/fermigas.page b/physics/quantum/fermigas.page new file mode 100644 index 0000000..0114b43 --- /dev/null +++ b/physics/quantum/fermigas.page @@ -0,0 +1,51 @@ +=============== +Fermi Gas +=============== + +Derivation of the Fermi Energy +--------------------------------- +Consider a crystal lattice with an electron gas as a 3 dimensional infinite +square well with dimensions :m:`$l_{x}, l_{y}, l_z$`. The wavefunctions of +individual fermions (pretending they are non-interacting) can be seperated +as :m:`$\psi(x,y)=\psi_{x}(x)\psi_{y}(y)\psi_{z}(z)$`. The solutions will be +the usual ones to the Schrodinger equation: + +:m:`$$\frac{-\hbar^2}{2m}\frac{d^2 \psi_x}{dx}=E_x \psi_x$$` + +with the usual wave numbers :m:`$k_x=\frac{\sqrt{2mE_x}}{\hbar}$`, and quantum +numbers satisfying the boundry conditions :m:`$k_x l_x = n_x \pi$`. The full +wavefunction for each particle will be: + +:m:`$$\psi_{n_{x}n_{y}n_{z}}(x,y,z)=\sqrt{\frac{4}{l_{x}l_{y}}}\sin\left(\frac{n_{x}\pi}{l_{x}}x\right)\sin\left(\frac{n_{y}\pi}{l_{y}}y\right)\sin\left(\frac{n_{z}\pi}{l_{z}}z\right)$$` + +and the associated energies (with :m:`$E = E_x + E_y + E_z$`): + +:m:`$$E_{n_{x}n_{y}n_z}=\frac{\hbar^{2}\pi^{2}}{2m}\left(\frac{n_{x}^{2}}{l_{x}^{2}}+\frac{n_{y}^{2}}{l_{y}^{2}}+\frac{n_{z}^{2}}{l_{z}^{2}}\right)=\frac{\hbar^2|\vec{k}|^2}{2m}$$` + +where :m:`$|\vec{k}|^2$` is the magnitude of the particle's k-vector in k-space. +This k-space can be imagined as a grid of blocks, each representing a possible +particle state (with a double degeneracy for spin). Positions on this grid have +coordinates :m:`$(k_{x},k_{y},k_z)$` corresponding to the positive integer +quantum numbers. These blocks will be filled +from the lowest energy upwards: for large numbers of occupying particles, +the filling pattern can be approximated as an expanding spherical shell with +radius :m:`$|\vec{k_F}|^2$`. + +Note that we're "over counting" the number of occupied states because the +"sides" of the quarter sphere in k-space (where one of the associated quantum +numbers is zero) do not represent valid states. These surfaces can be ignored +for very large N because the surface area to volume ratio is so low, but the +correction can be important. There will then be a second correction due to +removing the states along the individual axes twice (once for each +side-surface), u.s.w. + +The surface of this shell is called the Fermi surface and represents the most +excited states in the gas. The radius can be derived by calculating the total +volume enclosed: each block has volume :m:`$\frac{\pi^3}{l_x l_y +l_z}=\frac{\pi^3}{V}$` and there are N/2 blocks occupied by N fermions, so: + +:m:`$$\frac{1}{8}(\frac{4\pi}{3} |k_{F}|^{3})&=&\frac{Nq}{2}(\frac{\pi^{3}}{V})\\|k_{F}|&=&\sqrt{\frac{3Nq\pi^2}{V}}^3=\sqrt{3\pi^2\rho}^3$$` + +:m:`$\rho$` is the "free fermion density". The corresponding energy is: + +:m:`$$E_{F}=\frac{\hbar^{2}}{2m}|k_{F}|^{2}=\frac{\hbar^{2}}{2m}(3\rho \pi)^{2/3}$$` diff --git a/physics/special relativity b/physics/special relativity deleted file mode 100644 index 37fd3e9..0000000 --- a/physics/special relativity +++ /dev/null @@ -1,56 +0,0 @@ -=========================== -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..37fd3e9 --- /dev/null +++ b/physics/special relativity.page @@ -0,0 +1,56 @@ +=========================== +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/units b/physics/units deleted file mode 100644 index b1968f4..0000000 --- a/physics/units +++ /dev/null @@ -1,53 +0,0 @@ -====================== -Units -====================== - -.. contents:: - -SI Units --------------------- -The SI system uses meters-kilograms-seconds. It also defines the Coulomb as -a unit for measuring electric charge, which introduces redundant conversions -between mass-length-time units and the electric charge. - -cgs Units --------------------- -The cgs system uses centimeters-grams-seconds, and also defines electric charge -in terms of the fundamental quantities of mass, length, and time. The unit of -charge is "esu" or electrostatic unit. - -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 -(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 -these three constants. See _`Plank Units` for more on using G as a unit. - -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:`$[Q]=[gm]^{a}[cm]^{b}[sec]^{c}=[c]^{\alpha}[\hbar]^{\beta}[eV]^{\gamma}$`: - -:m:`$$(\alpha,\beta,\gamma)=\left(\begin{array}{ccc} -2 & 1 & 0\\ 0 & 1 & 1\\ 1 & -1 & -1\end{array}\right)\left(\begin{array}{c} a\\ b\\ c\end{array}\right)=(-2a+b,b+c,a-b-c)$$` - -or in reverse: - -:m:`$$(a,b,c)=\left(\begin{array}{ccc} 0 & 1 & 1\\ 1 & 2 & 2\\ -1 & -1 & -2\end{array}\right)\left(\begin{array}{c} \alpha\\ \beta\\ \gamma\end{array}\right)=(\beta+\gamma,\alpha+2\beta+\gamma,-\alpha-\beta-2\gamma)$$` - -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 -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 -important (similar to how the speed of light and Plank's constant provide -estimation of when special relativistic and quantum mechanical effects become -important). - - diff --git a/physics/units.page b/physics/units.page new file mode 100644 index 0000000..b1968f4 --- /dev/null +++ b/physics/units.page @@ -0,0 +1,53 @@ +====================== +Units +====================== + +.. contents:: + +SI Units +-------------------- +The SI system uses meters-kilograms-seconds. It also defines the Coulomb as +a unit for measuring electric charge, which introduces redundant conversions +between mass-length-time units and the electric charge. + +cgs Units +-------------------- +The cgs system uses centimeters-grams-seconds, and also defines electric charge +in terms of the fundamental quantities of mass, length, and time. The unit of +charge is "esu" or electrostatic unit. + +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 +(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 +these three constants. See _`Plank Units` for more on using G as a unit. + +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:`$[Q]=[gm]^{a}[cm]^{b}[sec]^{c}=[c]^{\alpha}[\hbar]^{\beta}[eV]^{\gamma}$`: + +:m:`$$(\alpha,\beta,\gamma)=\left(\begin{array}{ccc} -2 & 1 & 0\\ 0 & 1 & 1\\ 1 & -1 & -1\end{array}\right)\left(\begin{array}{c} a\\ b\\ c\end{array}\right)=(-2a+b,b+c,a-b-c)$$` + +or in reverse: + +:m:`$$(a,b,c)=\left(\begin{array}{ccc} 0 & 1 & 1\\ 1 & 2 & 2\\ -1 & -1 & -2\end{array}\right)\left(\begin{array}{c} \alpha\\ \beta\\ \gamma\end{array}\right)=(\beta+\gamma,\alpha+2\beta+\gamma,-\alpha-\beta-2\gamma)$$` + +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 +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 +important (similar to how the speed of light and Plank's constant provide +estimation of when special relativistic and quantum mechanical effects become +important). + + -- cgit v1.2.3