Easy calculations can show that the constancy of the speed of light is preserved:
A light signal is sent in a direction XY-X'Y' from their common position, when O and O' coincide at t=t'=0, forming an angle b with the X axis. If O measures the speed of light with components Ux = c.cos b, Uy = c.sin b and Uz = 0, then O' will measure:
U'x = (c.cos b - v.sin a)/ [1 - 2.v/c.sin(a + b) + v²/c²]1/2
U'y = (c.sin b - v.cos a)/ [1 - 2.v/c.sin(a + b) + v²/c²]1/2
U' = [(U'x)² + (U'y)²]1/2 = [c².(1 - 2.v/c.sin(a + b) + v²/c²)/(1 - 2.v/c.sin(a + b) + v²/c²)]1/2 = c
Lorentz transformation applied to curvilinear motion in three dimensions.
By doing a similar procedure we can obtain the Lorentz Transformation for a three dimensional motion of the moving coordinate system O' (always considering the speed of light constant for any observer). The results are the following ones:
dx' = (dx-v.dt.cos b .sin a)/(1-v²/c²)1/2
dy' = (dy-v.dt.cos b .cos a)/(1-v²/c²)1/2
dz' = (dz-v.dt.sin b)/(1-v²/c²)1/2
dt' = [(cos b .sin a.dt - v.dx/c²)² + (cos b .cos a.dt - v.dy/c²)² + (sin b .dt - v.dz/c²)²]1/2/(1-v²/c²)1/2
where the angle a has the same meaning as it was defined before in the plane XY and the angle b is that formed by the direction of the velocity v with the perpendicular to the plane XY.
The Lorentz transformations for velocities are:
V'x = (Vx - v.cos b .sin a)/[(cos b .sin a.dt - v.dx/c²)² + (cos b .cos a.dt - v.dy/c²)² + (sin b .dt - v.dz/c²)²]1/2
V'y = (Vy - v.cos b .cos a)/[(cos b .sin a.dt - v.dx/c²)² + (cos b .cos a.dt - v.dy/c²)² + (sin b .dt - v.dz/c²)²]1/2
V'z = (Vz - v.sin b )/[(cos b .sin a.dt - v.dx/c²)² + (cos b .cos a.dt - v.dy/c²)² + (sin b .dt - v.dz/c²)²]1/2
Additionally to the angle q, there is an angle f formed by the radius r with the plane XY, where the following relationships are met:
v.dt.cos b .sin a = d(Ro - r.cos f .cos q)
v.dt.cos b .cos a = d( r.cos f .sin q)
v.dt.sin b = d(r.sin f)
In this case, as it is expected, the constancy of speed of light is also preserved.
At this moment, we have all the sets of equations for each case of Lorentz Transformations. As we will see later, these transformations will allow us to face the general theory of relativity, or said in simple words, the consideration of the Lorentz transformation for mass in relativity.
New definition of mass.
As we know, Lorentz Transformation starts with the statement that the speed of light is the same measured by different observers. This comes out from the fact that photons do not have rest mass. This behavior of light in the universe is a trascendent fact. When we try with other particles or bodies in the universe that do have rest mass, the situation is not the same; for example: Let's consider that one observer on the earth tries to measure the earth momentum, and that another observer fixed at a point located on the path described by the earth around the Sun and at a distance Ro from the center of mass of the earh and the sun, does the same. Let's consider fixed the center of mass of the system. Momentum measured by the observer located on the earth, becomes a zero value, because earth does not move relative to him. The other fixed observer, however, does measure a value for the momentum because earth moves relative to him. The fact, that measurements of physical magnitudes do not have any relationship between them for different observers, as it was shown with this simple example, indicates that we should look for a way to make consistent relations among physical magnitudes, when these are referred to particles with non zero mass rest. This will be the next task in this work.
A first natural way to find permanent relationships between physical magnitudes measured by two observers located in different reference systems is that both observers measure such magnitudes relative to the same point, each one with his own clock. In the case that we are considering, a natural point to take as their reference for both observers in their measurements is the center of mass of the system. The powerfulness of choosing this point can be noticed by taking the solar system as a more complex example for making these relations, in which the center of mass is the unique point that can be considered fixed in all this system.
Let's state the following situation, in order to apply the previously obtained curvilinear transformations: Let's consider that observers are fixed to each origin of coordinates, one of them on earth, moving reference system O', and the other at a point on the earth path, reference system O, fixed at a distance Ro from the center of mass, also fixed. And both measuring physical magnitudes of the movement of the earth around the sun, relative to the center of mass of this system, each one with his own clock and with his own measurement instruments (say, there are only two sets of instruments: those used by each observer). The angles a, b, f and q have the same meaning previously defined.
The fixed observer at O, will measure, relative to the center of mass, for two or three dimensions, the following values:
dx = Vx.dt = (v.cos b .sin a).dt;
dy = Vy.dt = (v.cos b .cos a).dt;
dz = Vz.dt = (v.sin b).dt;
If we substitute these values in the expression of time t', measured by the observer at O', in the moving reference system (earth), for two or three dimensions, according to the previously encountered relationships, we obtain:
dt' = [(cos b .sin a.dt - v.dx/c²)² + (cos b .cos a.dt - v.dy/c²)² + (sin b .dt - v.dz/c²)²]1/2/(1-v²/c²)1/2 = dt.(1-v²/c²)1/2
Say, in both cases the obtained relationship between both time differentials, dt' and dt, is simpler, and the same.
The distance measured by the observer at O', on earth, relative to the center of mass of the system, is given by what he properly measures (zero distance because he is on earth) minus the distance to the center of mass of the system. By putting the differentials of distance in terms of O, using the previously obtained transformations, we have:
Obviously dx-v.dt.sen a = 0, according to our last results, but wait, what is inside this zero is useful. Instead, let's put v.dt.sen a = d(Ro - r.cos q). Because Ro is a constant, we finally obtain the general simplification for the distance measured relative to the center of mass:
By dividing the differentials of distance by the differential of time, measured all of them relative to the center of mass, by each observer, we obtain the transformation for velocities:
So, we have demonstrated for two dimensions that if the reference of measurements, taken independently for both observers, is the center of mass of the system, the differentials of the components of distances or the proper differentials of distances, become relationed by a Lorentz factor. The same was done for velocities. This behavior also repeats and are the same for three dimensions (make the exercise!). In sum:
dX'cm = dx/(1-v²/c²)1/2 = (v.cos b .sin a).dt/(1-v²/c²)1/2
dY'cm = dy/(1-v²/c²)1/2 = (v.cos b .cos a).dt/(1-v²/c²)1/2
dZ'cm = dz/(1-v²/c²)1/2 = (v.sin b .dt/(1-v²/c²)1/2
dr' = dr/(1-v²/c²)1/2
From above we can observe that V'x = Vx/(1-v²/c²), in which we can establish that v'.sin a' = (v.sin a)/(1-v²/c²), but because v' = v/(1-v²/c²), then, sin a' = sin a, which implies, in this case, that angle a is invariant to the Lorentz Transformation, or that both observers measure the same angle a. Well, this is also true for the angles b, q and f. This fact was the expected, because angles are the relation of two distances (arc and radius) and the Lorentz factor is the same for both distances, cancelling out their values.
It can be demonstrated that not only dr' = dr/(1-v²/c²)1/2, but also r' = r/(1-v²/c²)1/2. As a check, we know that in general, v = [(dr/dt)² + (w.r)²]1/2, and that v' = v/(1-v²/c²). By noticing that v' = [(dr'/dt')² + (w'.r')²]1/2, where w' = d(q')/dt' = [d(q)/dt]/(1-v²/c²)1/2 = w/(1-v²/c²)¹/², it follows that r' = r/(1-v²/c²)1/2. From here, we then can write:
w' = w/(1-v²/c²)1/2
r' = r/(1-v²/c²)1/2
The fact that dr' = dr/(1-v²/c²)1/2 and r' = r/(1-v²/c²)1/2 implies that the Lorentz factor acts as it were a factor independent of the integration or derivation processes. This seems to be an interesting property of the Lorentz factor. This property will be very useful and will be used after, to simplify calculations.
With these results we are very close to establish the new definition of mass.
Let's go now, and again, to the movement of earth around the sun, which is not actually around the sun, but around the center of mass of the system of these two bodies. Let's say that in a solid mass, particles are firmly joined together and the angular momentum of an external force over the constituent particles of that solid mass compensates, and has a null effect on the rotational momentum of the solid mass. Given this character, the spin or the rotational momentum of a solid mass under the field of a gravitational force, is preserved. This noticeable effect of the gravitational field can be clearly viewed through the expression of the orbital momentum, on which we will concentrate our development.
In the case of a two solid mass isolated system the orbital momentum is constant in any reference system (for any observer). This is a consecuence of the general accepted law in physics about the preservation of angular momentum in an asolated system.
The value of the orbital momentum depends upon the reference point around the measurements are done. Observe that in the case of the spin of a solid body, rotational angular momentum always is taken relative to the center of mass of the body, and is the same for any observer. Similarly, if the point of reference for measurements of the orbital momentum is taken relative to the center of mass of the two solid masses, then either the spin or the orbital momentum should be independent of the observers, and the same for all of them. Let's try to demonstrate that the orbital momenta measured by any observers (each one with his own clock and with his own measurement instruments), relative to the center of mass of the whole system in any coordinate system, moving or fixed, are the same.
Let's continue with our isolated system of two masses in which the fixed observer, located at a point on the path of the earth, measures a constant orbital momentum of the earth given by L = m.r².w = Mo.(Ro)².Wo.
Similarly and independently, the moving observer located on the earth must measure, relative to the center of mass of the system, also a constant orbital momentum of earth given by L' = m'.r'².w' = Mo'.(Ro')².Wo'.
If we put the obtained expression of what is measured by the moving observer at O', in terms of what is measured by the fixed observer at O, according to the last previously encountered relationships, we have:
L' = m'.r'².w' = m'.[w/(1-v²/c²)1/2].[r²/(1-v²/c²)] = Constant
This relationship is very important. For instance, if earth stops at some instant (this is only an example), then at that instant v = 0, and both observers must measure the same angular momentum. In fact, evaluating the previous equation for this value of v = 0, we obtain, at this instant, L = L' as it is expected.
But, it is important to notice that the only way for the term at right of the equation being constant, or not affected by the variation of earth's velocity v, is that the earth mass vary in the following manner:
m' = m.(1-v²/c²)3/2
The mass with this expression cancels out all the variations of velocity v, in the whole right term, always!. Say,
L' = m'.r'².w' = m.(1-v²/c²)3/2.[w/(1-v²/c²)1/2].[r²/(1-v²/c²)] = m.r².w = L = Constant
The previous result also indicates that the angular momenta measured by both observers are the same, which is what we were trying to demonstrate. So, the orbital angular momentum measured in this way can also be considered as an inherent property of the solid mass in the system, and has the same value for any observer.
Also, the relationship between both measured masses, that originates the equality of measured momenta, is m' = m.(1-v²/c²)3/2. But, the measured mass by the observer located at O', m', neccesarily must be equal to the rest mass Mo, because the mass m' is fixed relative to the observer at O'.
Thus, we have obtained three main results in this first part of this work:
1) Lorentz factor seems to be independent of the integration and derivation processes between both observers, when there are other magnitudes that vary and depend on measurements, and measurements are taken referred to the center of mass of the system. We will come on this feature in next Parts of this work.
2) Orbital Angular Momentum is invariant to the Lorentz Transformation when measurements are taken referred to the center of mass of the system.
3) The moving mass measured by an observer located on it, is the rest mass Mo for this observer. Thus, the mass viewed in motion, relative to the center of mass of the system by the fixed observer, m, is related to the rest mass Mo (which is absolutely unique for both observers) as:
m = Mo/(1-v²/c²)3/2
or said in simple words: The fixed observer measures a variation in the mass of the earth (which moves relative to him) that depends on its velocity according to the previously encountered relationship, where Mo is the rest mass at v = 0.
This is the referred "new" definition of mass, different of that of Einsten's. I was atonished with this result!. However, also Einstein found it and called it LONGITUDINAL MASS, in his work on the "Slowly Accelerated Electron". This is the reason because we put quotations on the word "new". But, anyway, we have encountered it in a different manner as he did. Actually, I don't know why he discarded it in his postwork, without any convincible explanation!.
It is important to notice that all our development is based only upon the principle of the constancy of the speed of light. No other principle has been used: neither any principle of equivalence nor any relativity principle. It means that, it's very easy to disagree this procedure and also to beat it, according to an Einsteinian optic (and reality). But, the challenge is there and we accept it. We think that, it is not necessary any other assumption and we will try to maintain this line of thinking along all this work.
Now the problem for us is to check if this new definition works. We will see in the next Part 2, that it really works!.
For that, we need to develope the new definitions of Energy and Linear Momentum (or just Momentum) and see if they reduce to the classical Newtonian expressions, at low velocities compared with that of light, as it is expected. This will be our next task.
Get your own Free Home Page