The last course report covered the first six classes of the relativity unit. This week, we had the final two relativity lectures, and today was the start of quantum mechanics.
Class 7: This lecture was about how you can use special relativity to show that a magnetic field in a stationary frame is an electric field in a moving frame. The basic idea is that when you move to a frame that is moving in the same direction as the (canonical) current, you see the spacing between the negative charges decrease due to length contraction, meaning that the wire no longer appears neutral. This leads to an attractive force on a stationary charge in that frame, that turns out to be equal to the magnetic force experienced by the charge in the lab frame.
I saw this as an undergrad taking E&M out of Purcell, and it's one of the few things I remember clearly from that class (the professor was approximately eight hundred years old, and his handwriting was so bad he made every Greek letter look like a "q" (or a "Q")). The textbook I'm using contains a brief and equation-free discussion of the topic; I fleshed it out with equations from another source, and it makes a good way to tie up all the special relativity topics we covered.
Class 8: I had meant to use this class as a math background class, doing the Swashbuckling Physicist's Introduction to Complex Exponentials, because the math department doesn't teach them in the classes that are pre-requisites for my class. In the comments to the last report, though, dr. dave asked "No mention of General Relativity?" I realized that that was a serious omission, and the math topic isn't really essential for the next week or so of classes, so I decided to put that off, and do one lecture on General Relativity.
The class was essentially identical to what I told the dog. Only, you know, without the dog. I talked about the equivalence principle and how that leads to the conclusion that light must follow a curved path in a gravitational field. And if you put that together with the fact that light always follows the shortest path between two point, that means that gravity must cause space-time to curve. I ended by talking a little about the Eddington eclipse observation (and the epic failures that preceded it), gravitational lensing, and LIGO, which wrapped things up nicely, because we started relativity with Michelson-Morley.
Thursday, I gave an exam during the lab period. The test was designed to fit in our 65-minute lecture periods, but as usual, the average time to completion was more like an hour and twenty minutes. Which is why I give the tests in the lab period.
I haven't graded them yet, and won't talk about the grades here, anyway.
Class 9: Today's class was the beginning of quantum mechanics, dealing with black-body radiation. I did a quick review of Young's double slit experiment, which shows that light is a wave, and then talked about the observed properties of thermal radiation (Wien displacement law, Stefan-Boltzmann law, Planck formula for the spectrum). To illustrate the problem Planck faced while trying to derive his formula, I went through the Rayleigh-Jeans approach, leading to the Ultraviolet Catastrophe (which I realize is slightly ahistorical, but it gets the point across), and then explained how Planck's quantum postulate fixes the problem.
I didn't go through the math of how to get the formula from that assumption, because it's not critical to the class. I did go through showing how the Planck formula recovers the Rayleigh-Jeans result in the long wavelength limit, and how you can get the Wien displacement law from the Planck formula. The Stefan-Boltzmann law is part of the homework assignment.
It was a pretty good class, especially considering that I spent the morning distracted by worrying about Sick Baby. I ran about two minutes over, which is good for me, and got through everything I wanted to cover.
Monday is the photoelectric effect, then the Compton effect. Which are also the two labs we'll be doing in the next few weeks, so that's good...
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Would you mind elaborating on "The test was designed to fit in our 65-minute lecture periods, but as usual, the average time to completion was more like an hour and twenty minutes"? I've been reading about how to figure out how long it should take an undergraduate to finish an exam, and this is something I haven't really seen addressed.
Here is the paradigmatic unification of electricity and
magnetism via Einstein's Special Theory Relativity, by
a pair of sophisticated Establishment authors from
Oslo and Halifax. Their advanced textbook really is
aimed at Cosmologists, with subtle mathematical
Physics on such topics as Superspace, spherical shell
of dust in a vaccum, five-dimensional brane cosmology,
the Randall-Sundrum models, and Kaluza-Klein
cosmology.
I assure you that these authors have been exceedingly careful to state their assumptions (of which they are conscious) in a way that is utterly acceptable to The Establishment, and that their math is self-consistent and free of error (in and of itself).
Ultimately, the Establishment will only publish
challenges to the paradigm which are couched in
exactly this kind of notation and argument, if they
publish them at all. To the extent that we who teach Physics sometimes get people with weird anti-Relativity opinions (new Aether believers, Michaelson-Morley denialists, those who psychiatrically attack Einstein the way that Intelligent Designers psychiatrically attack Darwin) I tell them that if they are unable or unwilling to play the game by Establishment rules, they have differing agendae. But I do ask them to consider the below carefully.
==============================
Einstein's General Theory of Relativity
With Modern Applications in Cosmology
Oyvind Gron, and Sigbjorn Hervik
Springer
2007, XX, 538 p., Hardcover
ISBN: 978-0-387-69199-2
$109.0
2.14 (pp.40-42) [with ASCII used in this blog posting to
indicate the properly typeset and numbered equations
in the textbook]
==============================
Magnetism as a relativistic second-order effect
Electricity and magnetism are described completely by
Maxwell's equations of the electromagnetic field.
[they then display Heaviside's equations]
together with Lorentz's force-law
F = q(E + vXB).
However, the relation between the magnetic and the
electric force was not fully understood until Einstein
had constructed the special theory of relativity. Only
then could one clearly see the relationship between
the magnetic force on a charge moving near a current
carrying wire and the electric force between charges.
We shall consider a simple model of a current carrying
wire in which we assume that the positive ions are at
rest while the conducting electrons move with the
velocity v. The charge per unit length for each type
of charged particle is
lambda_hat = Sne
where S is the cross-sectional area of the wire, n the
number of particles of one type per unit length, and e
the charge of one particle. The current in the wire
is
J = Snev = lambda_hat v
The wire is at rest in an inertial frame CapSigma_hat.
As observed in CapSigma_hat it is electrically
neutral. Let a charge q move with a velocity u along
the wire in the opposite direction of the electrons.
The rest frame of q is CapSigma. The wire will now be
described from CapSigma (see Fig. 2.16 and 2.17).
Note that the charge per unit length of the particles
as measured in their own rest frame CapSigma_0 is
lambda_0- = lambda_hat (1 - v^2/c^2)^(1/2)
lambda_0+ = lambda_hat
since the distance between the electrons is Lorentz
contracted in CapSigma_hat compared to their distances
in CapSigma_0.
The velocities of the particles measured in CapSigma
are
v- = -(v+u)/(1 + uv/c^2)
and
v+ = -u.
The charge per unit length of the negative particles
as measured in CapSigma, is
lambda- = (1 - (v-)^2/c^2))^(-1/2) lambda_0.
Substitution from Eq. (2.79) and (2.80) gives
lambda- = gamma (1 + uv/c^2))^(-1/2) lambda_hat
where gamma = (1 - u^2/c^2)^(-1/2).
In a similar manner, the charge per unit length of the
positive particles measured in CapSigma is found to be
lambda+ = gamma lambda_hat
Thus, as observed in the rest frame of q the wire has
a net charge per unit length
lambda = lambda- - lambda+
= (gamma u v/c^2) lambda_hat.
As a result of the different Lorentz contractions of
the positive and negative ions when we transform from
their respective rest frames to CapSigma, a current
carrying wire which is electrically neutral in the
laboratory frame, as observed to be electrically
charged in the rest frame of the charge q.
As observed in this rest frame there is a radial
electric field with field strength
E = lambda/(2 pi epsilon_0 r).
Then a force F acts on q, this is given by
F = qE = (q lambda)/(2 pi epsilon_0 r)
= (lambda_hat v)/(2 pi epsilon_0 c^2 r) gamma q u.
If a force acts upon q as observed in CapSigma_hat
then a force also acts on q as observed in CapSigma.
According to the relativistic transformation of a
force component in the same direction as the relative
velocity between CapSigma_hat and CapSigma, this force
is
F_hat = (gamma ^ (-1)) F
= (lambda_hat v)/(2 pi epsilon_0 c^2 r) q u.
Inserting
J = lambda_hat v
from equation (2.78) and using
c^2 = (epsilon_0 mu_0)^(-1)
(where mu_0 is the permeability of a vacuum) we obtain
F_hat = (mu_0 J / 2 pi r) q u.
This is exactly the expression obtained if we
calculate the magnetic flux-density B_hat around the
current carrying wire using Ampere's circuit law
B_hat = (mu_0 J / 2 pi r)
and use the force-law (Eq. (2.77)) for a charge moving
in a magnetic field
F_hat = q u B_hat.
We have seen how a magnetic force appears as a result
of an electrostatic force and the special theory of
relativity. The considerations above have also
demonstrated that a force which is identified as
electrostatic in one frame of reference is observed as
a magnetic force in another frame. In other words,
the electric and the magnetic force are really the
same. What an observer names it depends upon his
state of motion.
==============================
ooh! you mean I contributed something?! sweet.
I still find it a challenge to say anything coherent about general relativity to a class w/ limited math skillz. The past few years I've had students drawing circles and triangles on giant "punch balloons" to explore deviations from non-Euclidean geometry, and how they depend on the curvature of the surface. It's actually been pretty useful, I think. (I also constructed a 5 foot hyperbolic surface out of wood and duct tape so that they could make measurements on a negative-curvature surface as well!) This all leads into an incredibly hand-waving lecture that tries to get the point of the "metric tensor" across... with only the vaguest sort of success.
So your book follows an historical approach to QM? Will it take you the book equivalent of 15 years, wallowing in the errors of the Rutherford and Bohr models, before you get to actual quantum mechanics circa 1926? This seems to me to be the place most in need of reform in the undergrad curriculum. Why not start with the extremely real demonstration of matter waves that tells you all of classical mechanics is wrong, particularly since you can do the demonstration in your very own lab.
Question:
When you set your exam time, do you use a factor of four or three times your own detailed solution time? I used to time the setup of all of the problems separately from the computing of an answer, under the assumption that students were as fast or faster than I am on a calculator, but I now find that most of them are as slow with the calculator as they are with problem setup!
Would you mind elaborating on "The test was designed to fit in our 65-minute lecture periods, but as usual, the average time to completion was more like an hour and twenty minutes"? I've been reading about how to figure out how long it should take an undergraduate to finish an exam, and this is something I haven't really seen addressed.
I usually estimate that it will take students three times as long to solve the problems as it takes me to write out the detailed solutions. That's usually an underestimate, though.
I'm generally happy if the best students in the class finish in the time the test was designed for. I don't like to take papers away from people who are still working, so I'll let everybody go a bit longer than intended.
I still find it a challenge to say anything coherent about general relativity to a class w/ limited math skillz.
I probably wouldn't've said anything at all about it if I hadn't done that dog dialogue already. Sinc eI had already put in most of the work, I figured what the hell.
So your book follows an historical approach to QM? Will it take you the book equivalent of 15 years, wallowing in the errors of the Rutherford and Bohr models, before you get to actual quantum mechanics circa 1926? This seems to me to be the place most in need of reform in the undergrad curriculum. Why not start with the extremely real demonstration of matter waves that tells you all of classical mechanics is wrong, particularly since you can do the demonstration in your very own lab.
The textbook I'm using takes the historical approach, so I pretty much follow along. I speed things up quite a bit, though-- there are at most a half-dozen classes of the historical approach (black-body, photoelectric effect, Compton effect, Bohr model, de Broglie waves, Davisson-Germer experiment) before hitting them with the Schroedinger equation.
I think the historical approach works well as a first introduction, both because it sort of mirrors the process of coming to terms with quantum weirdness , and also because it provides some compelling narratives to build lectures around. And we are, after all, a liberal arts college, so it's good for them to get a little history with their science.
etailed solution time? I used to time the setup of all of the problems separately from the computing of an answer, under the assumption that students were as fast or faster than I am on a calculator, but I now find that most of them are as slow with the calculator as they are with problem setup!
As I said above, I estimate about a factor of three, though that does tend to be an underestimate. I'm not all that systematic about it, though, and often cop out a bit by constructing problems that parallel old exam questions.
As for the calculator issue, for this class especially, I deliberately try to construct problems that don't have numerical answers. I do this for two reasons: mostly because this is sort of the gateway to the upper-level majors classes, where numerical answers are much rarer, but also because they're easier to grade. Students these days have a maddening habit of plugging numbers in from the very first step of the problem, and then manipulating six-digit numbers (without units) for the rest of the problem, making it almost impossible to figure out where they went wrong. If the whole problem is symbolic algebra, it's much easier to find mistakes and assign partial credit.
And if you put that together with the fact that light always follows the shortest path between two point[s], that means that gravity must cause space-time to curve.
Of course, one might wonder why we should assume that light always follows the shortest path between two pointsâat least in the presence of gravity. Understanding why this is ultimately correct takes one right to the heart of general relativity, and the singular fact that its field equations dictate geodesic paths for free particles (neglecting the interaction of spin and spacetime curvature).
As Dr. Dave says, it's very hard to avoid resorting to mere hand-waving...
No Need for Einstein
Kepler (demolish) Vs Einstein's
Ending Einstein's space jail of time in 2009 that led to fraud Symbol E=mc²
Areal velocity is constant: r² θ' =h Kepler's Law
h = 2Ï a b/T; b=aâ (1-ε²); a = mean distance value; ε = eccentricity
r² θ'= h = S² w'
S = r exp (Ỡwt); h = [r² Exp (2iwt)] w'=r²θ'
w' = (θ') exp [-2(i wt)]
w'= (h/r²) [cosine 2(wt) - Ỡsine 2(wt)] = (h/r²) [1- 2sine² (wt) - Ỡsin 2(wt)]
w' = w'(x) + Ỡw'(y) ; w'(x) = (h/r²) [ 1- 2sine² (wt)]
Î w'= w'(x) â (h/r²) = - 2(h/r²) sine² (wt) = - 2(h/r²) (v/c) ² v/c=sine wt
(h/ r²)(Perihelion/Periastron)= [2Ïa.aâ (1-ε²)]/Ta² (1-ε) ²= [2Ïâ (1-ε²)]/T (1-ε) ²
Î w' = [w'(x) â h/r²] = -4Ï {[â (1-ε²)]/T (1-ε) ²} (v/c) ² radian per second
{x [180/Ï;degrees]x[100years=36526days;century]x[3600;seconds in degree]
Î w" = (-720x36526x3600/T) {[â (1-ε²]/(1-ε)²} (v/c)² seconds of arc per century
This Kepler's Equation solves all the problems Einstein and all physicists could not solve
DI Her Binary starts systems
The circumference of an ellipse: 2Ïa (1 - ε²/4 + 3/16(ε²)²- --.) â 2Ïa (1-ε²/4); R =a (1-ε²/4) v=â [G m M / (m + M) a (1-ε²/4)] â â [GM/a (1-ε²/4)]; m<>Exp (ì w t) ---------->> S=r Exp (ì wt) Nahhas' Equation
Orbit-------->> Orbit light sensing------>> Visual Orbit; Exp = Exponential
Particle ---->> light sensing of moving objects------------ >> Wave
Newton--------->>light sensing---------->> Quantum
Quantum = Newton x Visual Effects
Quantum - Newton = Relativistic = Optical Illusions
E (Energy by definition) = mv²/2 = mc²/2; if v = c
m = mass; v= speed; c= light speed; w= angular velocity; t= time
S = r Exp (ì w t) = r [cos (wt) + ì sin (wt)] Visual effects
P = visual velocity = change of visual location
P = d S/d t = v Exp (ì w t) + ì w r Exp (ì w t)
= (v + ì w r) Exp (ì w t) = v (1 + ì) Exp (ì w t) = visual speed; v = wr
E (visual energy= what you see in lab) = m p²/2; replace v by p in E = mv²/2
= m p²/2 = m v²/2 (1 + ì) ² Exp (2ì wt)
= mv²/2 (2ì) [cosine (2wt) + ì sine (2wt)]
=ì mv² [1 - 2 sine² (wt) + 2 ì sine (wt) cosine (wt)];v = speed; c = light speed
wt = Ï/2
E (visual) = ìmv² (1 - 2 + 0)
E (visual) = -ì mc² ⡠mc² (absolute value;-ì = negative complex unit) If v = c
w t = Ï/4
E (visual) = imv² [1-1 +á»] =-mc²; v = c
wt =-Ï/4+á»ln2/2; 2á» wt=-á»Ï/2 - ln2
Exp (2i wt) = Exp [-á»Ï/2] Exp [ln(1/2)]=[-á» (1/2)]
E (visual) = imv² (-á»/2) =1/2mc² v = c
Conclusion: E = mc² is the visual Illusion of E = mc²/2 joenahhas1958@yahoo.com. All rights reserved.
PS: In case of E=mc² claims to be rest energy claims then
E=1/2m (m v + m' r) ² = (1/2m) (m' r) ²; v = 0
E = (1/2m) (mc) ²; m' r =mc
E=mc²/2
Chad,
If my comment (#6) somehow elicited #7, I'm sorry. :)
(I truly hope you don't get any more like it.)
Einstein's Physics Dollar Store on Campus
MIT Harvard Cal-Tech
Sponsored by NASA
Why Relativity theory is not Physics and why Einstein's "thought" = 0
Walking the walk and talking the talk taking on all space-time confusion of physics by
MIT Harvard and Cal-Tech and all other Physics dollar stores departments
And why LHC burned itself
Visual Effects and the confusions of "Modern" physics
r --------- Light sensing of moving objects ------- S
Actual object----- Light --------- Visual object
r - -------cosine (wt) + i sine (wt) - S = r [cosine (wt) + i sine (wt)]
Newton-- Kepler's time visual effects -- Time dependent Newton
Particle -------------- Visual effects -------------------- Wave
Line of Sight: r cosine wt
r ------------------- r cosine (wt) line of sight light aberrations
A moving object with velocity v will be visualized by
light sensing through an angle (wt);w = constant and t= time
Also, sine wt = v/c; cosine wt = â [1-sine² (wt) = â [1-(v/c) ²]
A visual object moving with velocity v will be seen as S
S = r [cosine (wt) + i sine (wt)] = r Exp [i wt]; Exp = Exponential
S = r [â [1-(v/c) ²] + á» (v/c)] = S x + i S y
S x = Visual along the line of sight = r [â [1-(v/c) ²]
This Equation is special relativity length contraction formula
And it is just the visual effects caused by light aberrations of a
moving object along the line of sight.
In a right angled velocity triangle A B C: Angle A = wt; angle B = 90°; Angle C = 90° -wt
AB = hypotenuse = c; BC = opposite = v; CA= adjacent = c â [1-(v/c) ²]
Einstein's Physics Dollar Store on Campus
MIT Harvard Cal-Tech
Sponsored by NASA
Why Relativity theory is not Physics and why Einstein's "thought" = 0
Walking the walk and talking the talk taking on all space-time confusion of physics by
MIT Harvard and Cal-Tech and all other Physics dollar stores departments
And why LHC burned itself
Visual Effects and the confusions of "Modern" physics
r --------- Light sensing of moving objects ------- S
Actual object----- Light --------- Visual object
r - -------cosine (wt) + i sine (wt) - S = r [cosine (wt) + i sine (wt)]
Newton-- Kepler's time visual effects -- Time dependent Newton
Particle -------------- Visual effects -------------------- Wave
Line of Sight: r cosine wt
r ------------------- r cosine (wt) line of sight light aberrations
A moving object with velocity v will be visualized by
light sensing through an angle (wt);w = constant and t= time
Also, sine wt = v/c; cosine wt = â [1-sine² (wt) = â [1-(v/c) ²]
A visual object moving with velocity v will be seen as S
S = r [cosine (wt) + i sine (wt)] = r Exp [i wt]; Exp = Exponential
S = r [â [1-(v/c) ²] + á» (v/c)] = S x + i S y
S x = Visual along the line of sight = r [â [1-(v/c) ²]
This Equation is special relativity length contraction formula
And it is just the visual effects caused by light aberrations of a
moving object along the line of sight.
In a right angled velocity triangle A B C: Angle A = wt; angle B = 90°; Angle C = 90° -wt
AB = hypotenuse = c; BC = opposite = v; CA= adjacent = c â [1-(v/c) ²]
I, a Caltech graduate, would not spend an "Einstein's Physics Dollar" on someone rudely sticking leaflets on my car's windshield, or mumbling incoherently from atop a soapbox while I'm trying to hear the concert in the park, or who so deeply misunderstands the protocols of conversation in the blogosophere.
I think that we all know what light abberation is, know how to compute the hypotenuse of a right angled triangle, and know how to use complex exponentials.
We also know that Special Relativity is wrong, was admitted by Einstein to be wrong, and many of us have learned General Relativity.
So, Alexander Nahhas, we are not interested in your needing for psychiatric reasons to kill your father figure Einstein. Please go away and let the rest of us maintain polite conversations.
I can't believe I'm writing this, but Jonathan, please don't feed the trolls.
Sorry, Chad. I feel guilty that I may have left open the Gate to Hell when I posted my well-intentioned, properly cited, but far too long #2.
I like to think that I'm no more than 1/4 troll myself, but those genes make one prone to self-deception.
The Problem Cal-Tech and all others have no clue how to not solve by any physics
V1143Cgyni Apsidal motion solution
Introduction: For 350 years Physicists Astronomers and Mathematicians missed Kepler's time dependent equation that changed Newton's equation into a time dependent Newton's equation and together these two equations combine classical mechanics and quantum mechanics into one mechanics explains "relativistic" effects as the difference between time dependent measurements and time independent measurements of moving objects and solve all motion posted puzzles in all of Mechanics that Einstein and all 100,000 space-time "physicists" could not solve by space-time physics or any published physics.
All there is in the Universe is objects of mass m moving in space (x, y, z) at a location
r = r (x, y, z). The state of any object in the Universe can be expressed as the product
S = m r; State = mass x location:
P = d S/d t = m (d r/dt) + (dm/dt) r = Total moment
= change of location + change of mass
= m v + m' r; v = velocity = d r/d t; m' = mass change rate
F = d P/d t = d²S/dt² = Total force
= m(d²r/dt²) +2(dm/dt)(d r/d t) + (d²m/dt²)r
= mγ + 2m'v +m"r; γ = acceleration; m'' = mass acceleration rate
In polar coordinates system
r = r r(1) ;v = r' r(1) + r θ' θ(1) ; γ = (r" - rθ'²)r(1) + (2r'θ' + rθ")θ(1)
Proof:
r = r [cosθ î + sinθĴ] = r r (1); r (1) = cosθ î + sinθ Ĵ
v = d r/d t = r' r (1) + r d[r (1)]/d t = r' r (1) + r θ'[- sinθ î + cos θĴ] = r' r (1) + r θ' θ (1)
θ (1) = -sinθ î +cosθ Ĵ; r(1) = cosθî + sinθĴ
d [θ (1)]/d t= θ' [- cosθî - sinθĴ= - θ' r (1)
d [r (1)]/d t = θ' [ -sinθ'î + cosθ]Ĵ = θ' θ(1)
γ = d [r'r(1) + r θ' θ (1)] /d t = r" r(1) + r' d[r(1)]/d t + r' θ' r(1) + r θ" r(1) +r θ' d[θ(1)]/d t
γ = (r" - rθ'²) r(1) + (2r'θ' + r θ") θ(1)
F = m[(r"-rθ'²)r(1) + (2r'θ' + rθ")θ(1)] + 2m'[r'r(1) + rθ'θ(1)] + (m"r) r(1)
= [d²(mr)/dt² - (mr)θ'²]r(1) + (1/mr)[d(m²r²θ')/dt]θ(1) = [-GmM/r²]r(1)
d²(mr)/dt² - (mr)θ'² = -GmM/r² Newton's Gravitational Equation (1)
d(m²r²θ')/dt = 0 Central force law (2)
(2) : d(m²r²θ')/d t = 0 <==> m²r²θ' = [m²(θ,0)ϲ(0,t)][ r²(θ,0)ϲ(0,t)][θ'(θ, t)]
= [m²(θ,t)][r²(θ,t)][θ'(θ,t)]
= [m²(θ,0)][r²(θ,0)][θ'(θ,0)]
= [m²(θ,0)]h(θ,0);h(θ,0)=[r²(θ,0)][θ'(θ,0)]
= H (0, 0) = m² (0, 0) h (0, 0)
= m² (0, 0) r² (0, 0) θ'(0, 0)
m = m (θ, 0) Ï (0, t) = m (θ, 0) Exp [λ (m) + ì Ï (m)] t; Exp = Exponential
Ï (0, t) = Exp [ λ (m) + á» Ï (m)]t
r = r(θ,0) Ï(0, t) = r(θ,0) Exp [λ(r) + ì Ï(r)]t
Ï(0, t) = Exp [λ(r) + á» Ï (r)]t
θ'(θ, t) = {H(0, 0)/[m²(θ,0) r(θ,0)]}Exp{-2{[λ(m) + λ(r)]t + ì [Ï(m) + Ï(r)]t}} ------I
Kepler's time dependent equation that Physicists Astrophysicists and Mathematicians missed for 350 years that is going to demolish Einstein's space-jail of time
θ'(0,t) = θ'(0,0) Exp{-2{[λ(m) + λ(r)]t + á»[Ï(m) + Ï(r)]t}}
(1): d² (m r)/dt² - (m r) θ'² = -GmM/r² = -Gm³M/m²r²
d² (m r)/dt² - (m r) θ'² = -Gm³ (θ, 0) ϳ (0, t) M/ (m²r²)
Let m r =1/u
d (m r)/d t = -u'/u² = -(1/u²)(θ')d u/d θ = (- θ'/u²)d u/d θ = -H d u/d θ
d²(m r)/dt² = -Hθ'd²u/dθ² = - Hu²[d²u/dθ²]
-Hu² [d²u/dθ²] -(1/u)(Hu²)² = -Gm³(θ,0)ϳ(0,t)Mu²
[d²u/ dθ²] + u = Gm³(θ,0)ϳ(0,t)M/H²
t = 0; ϳ (0, 0) = 1
u = Gm³(θ,0)M/H² + Acosθ =Gm(θ,0)M(θ,0)/h²(θ,0)
mr = 1/u = 1/[Gm(θ,0)M(θ,0)/h(θ,0) + Acosθ]
= [h²/Gm(θ,0)M(θ,0)]/{1 + [Ah²/Gm(θ,0)M(θ,0)][cosθ]}
= [h²/Gm(θ,0)M(θ,0)]/(1 + εcosθ)
mr = [a(1-ε²)/(1+εcosθ)]m(θ,0)
r(θ,0) = [a(1-ε²)/(1+εcosθ)] m r = m(θ, t) r(θ, t)
= m(θ,0)Ï(0,t)r(θ,0)Ï(0,t)
r(θ,t) = [a(1-ε²)/(1+εcosθ)]{Exp[λ(r)+Ï(r)]t} Newton's time dependent Equation --------II
If λ (m) â 0 fixed mass and λ(r) â 0 fixed orbit; then
θ'(0,t) = θ'(0,0) Exp{-2ì[Ï(m) + Ï(r)]t}
r(θ, t) = r(θ,0) r(0,t) = [a(1-ε²)/(1+εcosθ)] Exp[i Ï (r)t]
m = m(θ,0) Exp[i Ï(m)t] = m(0,0) Exp [á» Ï(m) t] ; m(0,0)
θ'(0,t) = θ'(0, 0) Exp {-2ì[Ï(m) + Ï(r)]t}
θ'(0,0)=h(0,0)/r²(0,0)=2Ïab/Ta²(1-ε)²
= 2Ïa² [â (1-ε²)]/T a² (1-ε) ²; θ'(0, 0) = 2Ï [â (1-ε²)]/T (1-ε) ²
θ'(0,t) = {2Ï[â(1-ε²)]/T(1-ε)²}Exp{-2[Ï(m) + Ï(r)]t
θ'(0,t) = {2Ï[â(1-ε²)]/(1-ε)²}{cos 2[Ï(m) + Ï(r)]t - á» sin 2[Ï(m) + Ï(r)]t}
θ'(0,t) = θ'(0,0) {1- 2sin² [Ï(m) + Ï(r)]t - á» 2isin [Ï(m) + Ï(r)]t cos [Ï(m) + Ï(r)]t}
θ'(0,t) = θ'(0,0){1 - 2[sin Ï(m)t cos Ï(r)t + cos Ï(m) sin Ï(r) t]²}
- 2Ỡθ'(0, 0) sin [Ï (m) + Ï(r)] t cos [Ï (m) + Ï(r)] t
Πθ (0, t) = Real Πθ (0, t) + Imaginary Πθ (0.t)
Real Πθ (0, t) = θ'(0, 0) {1 - 2[sin Ï (m) t cos Ï(r) t + cos Ï (m)t sin Ï(r)t]²}
W(ob) = Real Πθ (0, t) - θ'(0, 0) = - 2 θ'(0, 0){(v°/c)â [1-(v*/c) ²] + (v*/c)â [1- (v°/c) ²]}²
v ° = spin velocity; v* = orbital velocity; v°/c = sin Ï (m)t; v*/c = cos Ï (r) t
v°/c << 1; (v°/c)² â 0; v*/c << 1; (v*/c)² â 0
W (ob) = - 2[2Ï â (1-ε²)/T (1-ε) ²] [(v° + v*)/c] ²
W (ob) = (- 4Ï /T) {[â (1-ε²)]/ (1-ε) ²} [(v° + v*)/c] ² radians
W (ob) = (-720/T) {[â (1-ε²)]/ (1-ε) ²} [(v° + v*)/c] ² degrees; Multiplication by 180/Ï
W° (ob) = (-720x36526/T) {[â (1-ε²)]/ (1-ε) ²} [(v°+ v*)/c] ² degrees/100 years
Wâ (ob) = (-720x26526x3600/T) {[â (1-ε²)]/ (1-ε) ²} [(v° + v*)/c] ² seconds /100 years
The circumference of an ellipse: 2Ïa (1 - ε²/4 + 3/16(ε²)²- --.) â 2Ïa (1-ε²/4); R =a (1-ε²/4)
v (m) = â [GM²/ (m + M) a (1-ε²/4)] â â [GM/a (1-ε²/4)]; m<