THE THEORY OF INVARIANCE
NGUYỄN GIANG THÀNH
While studying the Theory of Relativity, I came up with some thoughts which are presented in this booklet as the Theory of Invariance. Although the Ship Invariance is also sailed with two postulates of the Special Theory of Relativity, she will carry us to a distinctive world: The World of Invariance.
I hope that this beautiful world is not far away from the reality.
Thank you so much for taking a journey on the Ship Invariance,
Nguyễn Giang Thành I. Postulates Theory of Invariance is written based on two postulates: 1. The laws of physics are the same for all observers in uniform motion relative to one another. 2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light. Later, we will see how the theory named Theory of Invariance. I. II. Invariance of Simultaneity Experiment:
At two planets Alpha and Beta, both away from Earth 300,000 km, we install two light bulbs A and B. Dan is an observer who is standing still on Earth, and Lynn is on a spaceship flying from Alpha to Beta at a velocity v with respect to Earth. Clocks on Alpha, Earth, and Beta were synchronized. We are going to find what Dan sees.
As Alpha clock, Dan’s clock, and Beta clock display 0:00:01, light bulbs A and B are turned on. Photons from A and B begin to fly toward Earth. Lynn is also right above Dan.
As Alpha clock, Dan’s clock, and Beta clock display 0:00:016, Lynn passes near a third of distance from Earth to Beta. Photons from A and B pass near 2/3 of distances from Alpha and Beta to Earth.
3 As Alpha clock, Dan’s clock, and Beta clock display 0:00:02, photons from A and B approach Dan. Dan sees both light bulbs A and B flash simultaneously.
What happened to Lynn? Did she also see A and B flashed simultaneously?
At the time A and B were turned on, Lynn was right at above Dan. She was away from photons just emitted at A and B at the same distance L.
Call ∆tA the time period that photons from A need to approach her, then ∆tA = (L - 0)/c.
Call ∆tB the time period that photons from B need to approach her, then ∆tB = (L - 0)/c.
Hence, ∆tA = ∆tB.
Which means Lynn saw both light bulbs A and B flashed simultaneously. And because Dan also saw both light bulbs A and B flashed simultaneously, we can say that:
Two events that are simultaneous in one reference frame are simultaneous in another frame moving with respect to the first frame.
Hence, the space and time in this theory are invariant. This is the reason why the theory was named the Theory of Invariance.
Let us go over the experiment one more time: At the time Lynn and Dan saw the flashes, they were at different positions. This means one photon, at the same time, can be present at different positions in space, depending on what reference frames observers are in. However, frequencies of the photon are different with respect to observers in different reference frames. This frequency change is called Doppler Effect which will be evaluated in section V.
III. Blue Shift, Red Shift – Light under effect of Gravity
Experiment:
A photon with energy Eo is emitted at A in a uniform gravitational field g. AB = BC = h. Photon’s energy measured at B is E.
4 Energy E and height h are variable amounts. Hence, ratio E ⁄ Eo can be written as a function of height h:
E ⁄ Eo = f(h) and Eo ⁄ E = f(-h)
=> f(h). f(-h) = 1
We also have f(2h) = f(h).f(h)
Thus, f(h) = ah = ekh
kh E ⁄ Eo = e (3.1)
kh ∆E = E – Eo = Eo (e -1) (3.2)
The Equation (3.1) can also be interpreted as ratio of frequencies of the photon:
kh f ⁄ fo = e (3.3)
In real experiments, in cases of gh << c, ratio of frequency is measured:
2 f/fo ≈ 1 + gh/c (3.4)
Compare Equation (3.3) to Equation (3.4), we have k = g/c2. Equations (3.1), (3.2), and (3.3) become
gh/c² E ⁄ Eo = e (3.5)
gh/c² ∆E = Eo (e -1) (3.6)
gh/c² f ⁄ fo = e (3.7)
In case of a flash of light fo is emitted at a distance Ro from a planet mass M, radius R, then frequency of the flash on surface of the planet is
GM(Ro – R)/RoRc² f = foe (3.8)
In case of a flash of light fo is emitted on the surface of the planet, then frequency of the flash in infinity is
-GM/Rc² f∞= foe (3.9)
Consequently, no black hole exists. IV. Derivation of Mass – Energy Equivalence
Experiment:
An object mo is dropped freely from A to B in a uniform gravitational field g.
5 Call mo the rest mass, and mg the gravitational mass.
Gravitational mass mg and height h are variable amounts. Hence, we can write ratio of masses as a function of height h. mg/mo = f(h) and mo/mg = f(-h)
=> f(h). f(-h) = 1
We also have f(2h) = f(h). f(h)
Thus, f(h) = ah = ekh
kh mg = mof(h) = moe (4.1)
Call F the gravitational force, F = mgg.
By definition of work, we have work W
(4.2)
Now, we consider in case of the object mo “decays” into 2 “pieces” of flash of light right at the time it is dropped. One piece emits up and another emits down. The one emits up hit a mirror right away and it is reflected down along with the other one. Compare Equation gh/c² (4.2) to Equation (3.6), ∆E = Eo (e -1), we have k = g/c2 (4.3)
2 Eo = moc (4.4)
2 gh/c² W = moc (e -1) (4.5)
2 gh/c² Replace k = g/c into Equation (4.1), we have mg = moe (4.6)
gh/c² From Equation (3.5), E = Eoe , and Equation (4.4),
2 gh/c² => E = moc e . Then with Equation (4.6), we have
2 E = mgc (4.7)
V. Doppler Effect
Call v the velocity of an observer with respect to a light source.
Call fo the frequency of a flash of light with respect to the observer as vo = 0. Call f the frequency of a flash of light with respect to the observer as v = v. Frequency f and velocity v are variable amounts. Hence, we can write ratio of frequencies as a function of velocity v: f/fo = f(v) and fo/f = f(-v)
=> f(v). f(-v) = 1
6 We also have f(2v) = f(v). f(v)
=> f(v) = av = ekv (5.1)
In real experiments, in cases of v << c, ratio of frequencies is measured: f/fo = f(v) ≈ 1 ± v/c (5.2)
v/c Compare Equation (5.1) to Equation (5.2), we have k = 1/c. Thus, f = foe -v/c as light source and observer are approaching each other, and f = foe as they are receding from each other.
±v/c f = foe (5.3)
±v/c Equation (5.3) can be interpreted as E = Eo e . (5.4)
VI. Total Energy, Gravitational Mass, Kinetic Energy, Momentum
1. Total energy E:
Experiment:
2 Let an object mo (it’s Eo = moc ) “decay” into 2 “pieces” of flash of light while an observer is moving to the object mo with velocity v.
Apply Equation (5.4) into the experiment, the observer receives an amount of energy E
v/c -v/c 2 E = 0.5 Eo (e + e ). Then with Equation (4.4), Eo = moc , we have
2 v/c -v/c E = moc (e + e )/2
2 E = moc cosh (v/c) (6.1)
2. Gravitational mass mg:
2 From Equations (4.7), E = mgc , and (6.1)
=> mg = mo cosh (v/c) (6.2)
3. Kinetic energy KE:
KE = E - Eo
2 2 From Equations (4.4), Eo = moc , and (6.1), E = moc cosh (v/c)
2 => KE = moc [cosh (v/c) –1] (6.3)
4. Momentum P:
From the experiment above, we also have an amount of momentum of two pieces of flash light with respect to the observer is P
7 v/c -v/c P = 0.5 (Eo/c) (e – e )
P = moc sinh (v/c) (6.4)
2 From Equations (6.1), E = moc cosh (v/c), and (6.4)
2 2 2 2 4 2 2 => E – P c = mo c [cosh (v/c) – sinh (v/c)]
2 2 2 2 4 E = P c + mo c (6.5)
2 From Equations (6.1), E = moc cosh (v/c), and (6.4), we can define momentum P as
P = dE/dv (6.6) and from Equations (6.2), mg = mo cosh (v/c), and (6.4), we can define gravitational mass as mg = dP/dv. (6.7)
VII. Investigation of a free fall
Drop an object mo in a uniform gravitational field g from a height h. We are going to find velocity v of the object mo at the end of the fall:
2 gh/c² Apply the Law of Conservation of Energy into Equations (4.5), W = moc (e -1), and (6.3), we have: W = KE
2 gh/c² 2 => moc (e -1) = moc [cosh (v/c) –1]
=> cosh (v/c) = egh/c² (7.1)
v = c.cosh-1 (egh/c²) (7.2)
In case of mo falls freely toward a planet mass M from distance Ro to distance R v = c.cosh-1 (eGM(Ro – R)/RoRc²). (7.3)
VIII. Escape Velocity
From the Equation (7.3), v = c.cosh-1 (eGM(Ro – R)/RoRc²), we have escape velocity of an object launched from surface of a planet mass M, radius R, is ve
-1 GM/Rc² ve = c.cosh (e ) (8.1)
IX. Law of Universal Gravitation
The Law of Universal Gravitation state gravity force F
F = GMm/R2
8 F = mg
In the Theory of Invariance, m is gravitational mass mg, where, mg = mo cosh (v/c)
Because M and m have the same role, we modify gravity force F as
2 F = GMo cosh (vM/c). mo cosh (vm/c)/R (9.1)
Where,
Mo, mo are rest masses, vM, vm are velocities of M and m on their orbits around center of gravity of M and m, R is distance between M and m, G is gravitational constant.
We can write F = GMm/R2 as long as we know M and m are gravitational masses.
X. The Principle of Equivalence: Fact or myth?
From Equation (7.1), cosh (v/c) = egh/c², differentiate both sides, we have
Although acceleration is different from gravitation, absolute value of acceleration is just a little greater than absolute value of gravitation. This is the reason why they have been thought to be equivalent. XI. Investigation of Motion
1. Second law of motion
Call Fg the gravitational force, Fg = mgg, where, mg = mo cosh (v/c).
Fg = mo cosh (v/c) g
Apply Equation (10.1),
Fg = mo a
Call Fi the inertial force. Call mi the inertial mass, Fi = mia. Fi must be equal to Fg.
=> Fi = mia = Fg = mo a (11.1)
=> Inertial mass mi = mo (11.2)
2. Relation between momentum and inertial force
Combine Equation (11.2) with Equation (6.4), P = moc sinh (v/c), we have
9 P = mi v (11.4)
=> Fi v = P a (11.5)
3. Power
Power = dW/dt
2 From Equations Equation (6.3), KE = moc [cosh (v/c) – 1],
2 Power = d(moc cosh (v/c) – 1)/dt
Power = moc sinh (v/c) dv/dt
Power = mi v a (11.6)
Power = Fi v = P a
TABLE OF COMPARISON
Newton Theory of Relativity Theory of Invariance Mechanics Space & Time Invariant Relativistic Invariant Speed of light c c c Velocity unlimited v < c unlimited
Rest Mass m mo mo
Mass m
Gravitational mg = dP/dv mg mg = mi Mass mg = mo cosh (v/c)
Inertial Mass mi mi = mg
2 2 Rest Energy Eo = moc Eo = moc
10 E = m c2 E = mc2 g
2 Total Energy E = moc cosh (v/c)
2 2 2 2 2 E = P c + mo c 2 2 2 2 E =P c + moc
2 gh/c² Potential Energy PE = mgh PE = moc (e - 1)
2 Kinetic Energy KE = moc [cosh (v/c)-1]
P = m c sinh (v/c) P = mv o Momentum P = mv P = mi v P = dE/dv
Second Law of F = dP/dt F = dP/dt F = Pa / v
Motion F = ma F = mi a
Power Power = Fv Power = Fv = Pa Blue and Red f ≈ f (1 ± gh/c2) f ≈ f (1 ± gh/c2) f = f e±gh/c² Shift o o o
±v/c Doppler Effect f = fo e
Escape Velocity v = c cosh-1(eGM/Rc²) Black hole Radius R = 2GM/c2 R = 2GM/c2 No black hole exists F = GMm/R2 Law of Universal F = GMm/R2 Gravitation M and m are gravitational masses.
11 References: 1. Physics / Arthur Beiser. 5th edition, 1991 by Addison-Wesley Publishing Company, Inc. 2. Contemporary college physics / Edwin R. Jones and Richard L. Childers. 2nd edition, 1992 by Addison-Wesley Publishing Company, Inc. 3. Modern Physics / Stephen T. Thornton and Andrew Rex. 2nd edition, 2000 by Saunders College Publishing. 4. Calculus / Earl W. Swokowski, Michael Olinick, Dennis Pence, with the assistance of Jeffery A. Cole. 6th edition, 1994 by PWS Publishing Company. Speed of light c 2.99792458 x 108 m/s Gravitational constant G 6.67259 x 10-11 m3kg-1s-2 Acceleration of gravity g 9.80665 m/s2
30 Mass of Sun MS 1.99 x 10 kg
24 Mass of Earth ME 5.98 x 10 kg Radius of Earth (ave.) 6.38 x 106 m Radius of Earth’s orbit 1AU 1.50 x 1011 m
22 Mass of Moon MM 7.35 x 10 kg Radius of Moon 1.74 x 106 m Radius of Moon’s orbit 3.84 x 108 m π 3.141593 e 2.718282