This work consists of two parts. The first part: The Lorentz transformation has two derivations. One of the derivations can be found in the references at the end of the work in the “Appendix I” of the book marked by number one. The equations for this derivation [1]: The other derivation of the Lorentz transformation is the traditional hyperbolic equations: ; ; For these equations we found new equations: , . The second part: In the second part is the equation by which we derive Minkowski’s equation. It will be proved that Minkowski’s equation is the integral part of the Lorentz transformation.
The aim of this work is to express the hyperbolic equations by trigonometric equations. I would like to point out the simplicity of the used triangles by the derivation of the equations. This work is based on three equations. In the first part of this work we work with two equations:
New equation:
New equation:
We join them by the hyperbolic equation of Lorentz transformation. The new equations give the same results as the Lorentz transformation hyperbolic forms. For the derivation of the new equations high school mathematics is used. That’s the reason why the new equations are more transparent and easier to understand because they are simply. After this, we check several ways the correctness of the new equations. The new equations must be in accordance with Minkowsi’s equation. In the second part we study the t' equations which we derived in the first part. Here we focus on the case when x = ct. x = ct in the first part appearing equations is a special case. Here we introduce our third new equation:
We shall proof as well that the (
rentz transformation. Minkowski’s equation can be seen in the “Appendix I” of the book marked by number one in the references in [
Now come the traditional hyperbolic equations [
1)
2)
3)
4)
Instead of these I found new equations, and we shall derive them. First come the new equations, and then their derivations.
5)
6)
The aim is that in the Equations ((1) and (2)) appear in hyperbolic equations to express the trigonometric equations.
The triangle:
7) See
8) See
9) See
The hyperbolic function was introduced into the mathematics on the analogy of
The hyperbola simplest form is:
See
10)
11)
12)
13) Grey area (
(Note:
14) Course Equation (10), is identical with Equation (7). See Equations ((7) and (10)):
We shall express the (
First we prove that:
The
Now follow the derivation of equation
15) We substitute the Equation (14):
It is totally in accordance with the [
{See Equation (14)}:
16) See Equation (15):
17) See Equation (16):
We got the first new equation. We can see that the
The
We shall later substitute the Equation (17) into the Equations ((1) and (2)). But before it, we shall derive the new equation of (
In the Equation (14):
18) Now comes equation cotan
19) We substitute in the Equation (18), Equation (14):
Equation (16):
20) See Equation (19):
So we got our second new equation as well. That means that the
21) We substitute the Equation (17) and Equation (20) into the Equation (2):
Note: Equation (8):
Equation (13):
It is one of most important equation of this work. It is distinct that the both sides of equation give the same result. So the hyperbolic Equation (2), we can substitute by a trigonometric equation.
It is very important to prove that the Equations ((17) and (20)) are correct. Further we will prove the correctness of Equations ((17) and (20)) in several different ways.
Now we check the correctness of the Equation (21). So by using the new equations we derive the know equation of Lorentz transformation. So follows the known equation of x':
22) Equation (21):
See Equation (9):
23) So:
Form the Equation (21) we got exactly the known equation of Lorentz transformation. So that means that the Equation (21) is correct. Here it is clearly visible that the point of the Equation (21) is the simplicity.
For the determination of ct' the same procedure is used.
24) Now come the Equation (1):
25) We substitute the Equation (17) and Equation (20) into the Equation (1):
Now we check the correctness of the Equation (25). By the usage the new Equation (25) it is derived the Lorentz transformation other know equation t'.
Here we use the same procedure as above by the equation x'.
26) See Equation (25):
27) See Equation (7):
See Equation (9):
28) So:
From the Equation (25) we got exactly the known equation of Lorentz transformation. So that means that the Equation (25) is correct. Here it is clearly visible that the point of the Equation (25) is the simplicity.
Equation (21):
Equation (25):
Minkowsi’s equation:
So that means that the Equation (21) and Equation (25) are correct. It is visible that we got simply the well- known Minkowsi’s equation.
The aim of this derivation is to check the correctness of the Equations ((21) and (25)).
On the
29) This derivation needs the visible equations from the “Appendix I” of the book [
30) See
Equations (17): (
31) See
32) See
Equations (20): (
(
33) We substitute the Equation (30) (
It is visible that we got the same result with the equation (
So that means that the Equation (21),
In the references at the end of the work in the “Appendix I” of the book marked by number [
In the following derivation we measure the same light-ray from the standing and moving coordinate system. Exclusively in this case can be used that we substitute in the Equation (28) the x = ct equality. The x = ct equality substitution in the Equation (28) is given in [
So the travelled distance of the light-ray in the standing coordinate system is x = ct. Naturally in the movable coordinate system the travelled distance of the light-ray is x' = ct'.
From now on we shall study the travel distance of the light-ray in the standing and movable coordinate system. This gives the opportunity to derive Minkowski’s equation. From the view of the special relativity theorem it is very important to derive the Minkowski’s equation from the Lorentz transformation
So we take the Equation (28) and pick out the (t) variable. (Note: (x = ct); (x/c) = t)
34) See Equation (28):
35) Equation (t'/t):
36) The
For example, [
37) We substitute Equation (7): (
38) Course, the first member the Equation (37), is identical with Equation (35):
The Equation (38) can be found in [
The third member the Equation (37) can be found in [
In the introductory part we already mentioned, that it is very important that the equation
In further derivation we shall use the following equations from the Appendix I of the book [
39) The equations:
40) From the last equations we express the
First of all, we calculate again the “a” and “b” variables of the Lorentz transformation from
Then by our “a” and “b” equations we derive the
Now we continue the Equation (38).
Now we transform the Equation (38) so that we can determine the Lorentz transformation “a” and “b” variables.
41) We shall transform the Equation (38) and take the Equation (39) into account:
42) Taking in account the Equations ((41), and (30), (32)):
Equation (30):
Equation (32):
Note:
Equation (8):
Equation (13):
By this we proved that the
We mustn’t forget that we started from the fact that we measure from the standing and movable coordinate system the two oncoming light-rays. In the standing coordinate system, the travelled distance of the light-ray is x = ct. From here we started, we substituted it in the Equation (28) and could not get other result. If we look closer the Equation (25) and substitute the equation x = ct, it gives the same result.
At the Equation (41) we wrote the
The second member of the Equation (37):
43) So:
The following derivation aim is to prove that: (
Equation (39):
We express the
44) Variable
45) Variable
46) See Equations ((44) and (45)):
Equation (41):
47) See Equation ((46) and (41)):
48) We substitute Equation (46) (
Equation (43):
49) From the Equations ((47) and (48)), (
So we calculated “
The equations of “
See equations of (49) and equations of (40):
With this we derived Minkowsi’s equation. We proved that Minkowsi’s equation is the integral part of the Lorentz transformation. With that, we proved the legitimacy of
If Minkowski’s equation is expanded by coordinates “y” and “z”, then we get the four dimensional word logical equation [
The reason why the equation can be expanded by “y” and “z” coordinates comes from the fact that the derivation of the Lorentz transformation starts from that simplification that all actions happen on the “x” axe.
Equation (8):
Equation (13):
The equations ct':
Equations (25)-(28):
Equations (42):
{Condition:
The x' equations:
Equation (21):
We multiply all sides of the triangle on the
The OA distance on the
By this we showed that on the
We can draw the (ct + vt) dimension and
See
Equation (37):
Proof:
See
See
So, that means that the
Watching from the great speed moving coordinate system, the world seems to be deformed, while watching from the standing coordinate system the same world looks undeformed. But the same world cannot be deformed and undeformed at the same time. The higher the speed of a moving coordinate system, the more it seems to be deformed the world from there. This phenomenon is closer to the Doppler effect. So, if on the
Csizmadia Jozsef, (2016) The Equations of Lorentz Transformation. Journal of Modern Physics,07,952-963. doi: 10.4236/jmp.2016.79087