Einstein, Albert. Relativity: The Special and General Theory. Translated by Robert W. Lawson, Digireads.com Publishing, copyright 2012 (this edition)

Appendix I. Simple Derivation of the Lorentz Transformation (supplementary to section XI)

This essay is derived from Dr. Einstein's book on Relativity.

Professor Dr. Albert Einstein uses the Lorentz Transformation and his ideas about physics to develop Special Relativity. After looking at his derivation, I am not convinced of its veracity. I refer the interested reader to the source. What follows is my failure to make sense out of what he (or Dr. Lorentz) wrote.

In two co-ordinate systems, S and S', which are defined by axes (x,y,z) and (x',y',z'), at time zero the two systems coincide (are one). Actually S' is moving at speed v down the x-axis (S and S' coincide at t=0=t'). It is given that light is transmitted down the x-axis at t=0, which is also t'=0 for the x'-axis.

The formula for a light pulse down the x-axis is: x(t) = ct, where c=speed of light and t=time, especially in frame S. Einstein expresses the formula as x(t)-ct =0. (I note that the expression "x(t)" means that x is a function of t; ct means c times t; Einstein does not use the form x(t), preferring the more simple "x". Also, I use S in place of Einstein's K.) "Since the same light-signal has to be transmitted relative to" S' "with the velocity c, the propagation relative to the system" S' "will be represented by the analogous formula

x'-ct'=0." So Einstein has introduced us to two formulas, which I present as:

(1) x(t)-ct=0 and

(2) x'(t')-ct'=0. I am not sure why I bother differentiating between t and t'; they seem to be the same.

[I am reminded of a comedy sketch in which a man called Darrel introduces his two brothers, one called Darrel and the other called Darrel. -- from a Bob Newhart TV show, as I recall.]

So we have two formulas: x(t)-ct=0 and x'(t')-ct'=0 that were derived from light down the x-axis of S. (Just wondering, why don't I have c and c'; oh yes, the speed of light is a constant everywhere, so far.) "Those space-time points (events) which satisfy (1) must also satisfy (2)." (For some reason unclear to me, Einstein sets the formulas equal to one another but with a constant multiplier. That is, not "y=mx+b" but rather "y= n(mx+b)". Note, just as ct refers to c times t, mx refers to m times x, etc..) Maybe Einstein is generalizing the formula to points off the x-axis? So Einstein presents:

(x'-ct') = L(x-ct) [Actually, Einstein uses a lamda, not an L.] Equation 3

As a reminder, we are looking at 0=0, which is really 0=L0, which is yet 0=0. What are the constraints on L? Generally speaking, L cannot equal zero, although for "this" expression no harm is done if it were.

Now Einstein uses the word "similar". Mathematicians often use that word. He means do the same things we did before to get to this new, but similar, conclusion. Often there is no harm in its use. In fact, often its use helps to explain.

[I am reminded how Einstein hated using the word "Simultaneously" in his derivation of Special Relativity.]

"If we apply quite similar considerations to light rays which are being transmitted along the negative x-axis, we obtain the condition" (x'+ct') = M(x+ct) Equation 4 [I replaced Einstein's mu with an M.]

["But wait, before you go any father, will you love me forever?" from a song, sung by Meatloaf]

The comment about the L is applicable to the M. Let us continue.

So if we apply quite similar considerations to a light ray along the negative x-axis, shouldn't we obtain the exact same formula, i.e., Equation 3: (x'-ct') = L(x-ct) ? What's different and why is it different?

First of all, in Equation 4, isn't x' negative? Isn't c in Equation 4 going down the negative x-axis? And I have no idea how t' differs from t.

OK, let's review:

Equation 3: (x'(t')-ct') = L (x(t)-ct) : I put the t and t' back into the equation.

Equation 4: (x'(t')+ct') = M (x(t)+ct) : I put the t and t' back into the equation.

OK, I understand Equation 3, which by the way is a glorified 0=0 with L not equaling zero (for generality).

I don't like Equation 4. Let us rewrite it. Equation 4c: x'(t') - ct' = M( x(t) - ct). Hmm, that seems correct.

We, of course, know that x' is a negative quantity. Hmm, we need a complete makeover. "Powder!"

Equation 4 is another one of those zeros (perhaps in disguise, but it shouldn't be -- it's started outright to be zero).

OK, let's approach the problem using distances, instead of the naked Cartesian axes. First, I declare that all distances are positive. When I walk W meters forward and V meters back, neither distance variable is negative. Thus, I've walked a (W-V) distance from the start. Let's look at Equation 3 in abbreviated form (no t's shown).

From page 66, turning x-axis values of x into distances, d. If Special Relativity falls apart on a change of variables, there was't much to Special Relativity at all.

Deriving Equation 3d:

d=ct which leads to d-ct=0, Equation 1d. "Since the same light-signal has to be transmitted relative to K' with velocity c, the propagation relative to the system K' will be represented by the analogous formula." OK, I use S' for K', but Equation 2d is d'-ct'=0.

This is the Relativity equation. "Those space-time points (events) that satisfy (1)" [which is now 1d] "must also satisfy (2)." Equation 2d replaces Equation 2. "Obviously this will be the case when the relation" Equation 3d replacing Equation 3 "is fulfilled in general, where" L "indicated a constant; for, according to" (3d) that replaced (3), the disappearance of" (d-ct) which replaced (c-xt) "involves the disappearance of" (d'-ct') which replaced (x'-ct')"." Hmm, further restriction is that L is positive.

(1d): d-ct=0 : for S

(2d): d'-ct'=0 : for S'

So far, so good -- all I did is replace an "x" which may be positive or negative with a "d" which is only positive. No change here in formula here.

Paraphrasing, events which satisfy (1d) must also satisfy (2d). "Obviously" this occurs with this relation:

Equation 3d: (d'-ct')=L(x-ct)

I remind you Equation 3d is a cute way of saying 0=0. I add restrictions to L. L is greater than zero or less than zero. (L is never equal to zero.)

"If we apply quite similar considerations to light rays which are being transmitted along the negative x-axis, we obtain the condition" Comment: Here is where Einstein switches the signs in the equation. He creates (x'+ct')=M(x+ct), Equation 4. But I am using the positive quantity d', so my formula is (d'-ct')=M(d-ct).

Check out: http://www.physicsmyths.org.uk/print/lorentz.htm

"Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation"

OK, so Einstein's derivation is lacking; I thought so. And you don't have to take my word for it.

Unlike Dr. Lorentz and unlike Professor Dr. Einstein, I think there is a preferred reference frame for the speed of light! The preferred reference frame is called gravity. The local gravity provides the aether for light.

Appendix I. Simple Derivation of the Lorentz Transformation (supplementary to section XI)

This essay is derived from Dr. Einstein's book on Relativity.

Professor Dr. Albert Einstein uses the Lorentz Transformation and his ideas about physics to develop Special Relativity. After looking at his derivation, I am not convinced of its veracity. I refer the interested reader to the source. What follows is my failure to make sense out of what he (or Dr. Lorentz) wrote.

In two co-ordinate systems, S and S', which are defined by axes (x,y,z) and (x',y',z'), at time zero the two systems coincide (are one). Actually S' is moving at speed v down the x-axis (S and S' coincide at t=0=t'). It is given that light is transmitted down the x-axis at t=0, which is also t'=0 for the x'-axis.

The formula for a light pulse down the x-axis is: x(t) = ct, where c=speed of light and t=time, especially in frame S. Einstein expresses the formula as x(t)-ct =0. (I note that the expression "x(t)" means that x is a function of t; ct means c times t; Einstein does not use the form x(t), preferring the more simple "x". Also, I use S in place of Einstein's K.) "Since the same light-signal has to be transmitted relative to" S' "with the velocity c, the propagation relative to the system" S' "will be represented by the analogous formula

x'-ct'=0." So Einstein has introduced us to two formulas, which I present as:

(1) x(t)-ct=0 and

(2) x'(t')-ct'=0. I am not sure why I bother differentiating between t and t'; they seem to be the same.

[I am reminded of a comedy sketch in which a man called Darrel introduces his two brothers, one called Darrel and the other called Darrel. -- from a Bob Newhart TV show, as I recall.]

So we have two formulas: x(t)-ct=0 and x'(t')-ct'=0 that were derived from light down the x-axis of S. (Just wondering, why don't I have c and c'; oh yes, the speed of light is a constant everywhere, so far.) "Those space-time points (events) which satisfy (1) must also satisfy (2)." (For some reason unclear to me, Einstein sets the formulas equal to one another but with a constant multiplier. That is, not "y=mx+b" but rather "y= n(mx+b)". Note, just as ct refers to c times t, mx refers to m times x, etc..) Maybe Einstein is generalizing the formula to points off the x-axis? So Einstein presents:

(x'-ct') = L(x-ct) [Actually, Einstein uses a lamda, not an L.] Equation 3

As a reminder, we are looking at 0=0, which is really 0=L0, which is yet 0=0. What are the constraints on L? Generally speaking, L cannot equal zero, although for "this" expression no harm is done if it were.

Now Einstein uses the word "similar". Mathematicians often use that word. He means do the same things we did before to get to this new, but similar, conclusion. Often there is no harm in its use. In fact, often its use helps to explain.

[I am reminded how Einstein hated using the word "Simultaneously" in his derivation of Special Relativity.]

"If we apply quite similar considerations to light rays which are being transmitted along the negative x-axis, we obtain the condition" (x'+ct') = M(x+ct) Equation 4 [I replaced Einstein's mu with an M.]

["But wait, before you go any father, will you love me forever?" from a song, sung by Meatloaf]

The comment about the L is applicable to the M. Let us continue.

So if we apply quite similar considerations to a light ray along the negative x-axis, shouldn't we obtain the exact same formula, i.e., Equation 3: (x'-ct') = L(x-ct) ? What's different and why is it different?

First of all, in Equation 4, isn't x' negative? Isn't c in Equation 4 going down the negative x-axis? And I have no idea how t' differs from t.

OK, let's review:

Equation 3: (x'(t')-ct') = L (x(t)-ct) : I put the t and t' back into the equation.

Equation 4: (x'(t')+ct') = M (x(t)+ct) : I put the t and t' back into the equation.

OK, I understand Equation 3, which by the way is a glorified 0=0 with L not equaling zero (for generality).

I don't like Equation 4. Let us rewrite it. Equation 4c: x'(t') - ct' = M( x(t) - ct). Hmm, that seems correct.

We, of course, know that x' is a negative quantity. Hmm, we need a complete makeover. "Powder!"

Equation 4 is another one of those zeros (perhaps in disguise, but it shouldn't be -- it's started outright to be zero).

OK, let's approach the problem using distances, instead of the naked Cartesian axes. First, I declare that all distances are positive. When I walk W meters forward and V meters back, neither distance variable is negative. Thus, I've walked a (W-V) distance from the start. Let's look at Equation 3 in abbreviated form (no t's shown).

From page 66, turning x-axis values of x into distances, d. If Special Relativity falls apart on a change of variables, there was't much to Special Relativity at all.

Deriving Equation 3d:

d=ct which leads to d-ct=0, Equation 1d. "Since the same light-signal has to be transmitted relative to K' with velocity c, the propagation relative to the system K' will be represented by the analogous formula." OK, I use S' for K', but Equation 2d is d'-ct'=0.

This is the Relativity equation. "Those space-time points (events) that satisfy (1)" [which is now 1d] "must also satisfy (2)." Equation 2d replaces Equation 2. "Obviously this will be the case when the relation" Equation 3d replacing Equation 3 "is fulfilled in general, where" L "indicated a constant; for, according to" (3d) that replaced (3), the disappearance of" (d-ct) which replaced (c-xt) "involves the disappearance of" (d'-ct') which replaced (x'-ct')"." Hmm, further restriction is that L is positive.

(1d): d-ct=0 : for S

(2d): d'-ct'=0 : for S'

So far, so good -- all I did is replace an "x" which may be positive or negative with a "d" which is only positive. No change here in formula here.

Paraphrasing, events which satisfy (1d) must also satisfy (2d). "Obviously" this occurs with this relation:

Equation 3d: (d'-ct')=L(x-ct)

I remind you Equation 3d is a cute way of saying 0=0. I add restrictions to L. L is greater than zero or less than zero. (L is never equal to zero.)

"If we apply quite similar considerations to light rays which are being transmitted along the negative x-axis, we obtain the condition" Comment: Here is where Einstein switches the signs in the equation. He creates (x'+ct')=M(x+ct), Equation 4. But I am using the positive quantity d', so my formula is (d'-ct')=M(d-ct).

Check out: http://www.physicsmyths.org.uk/print/lorentz.htm

"Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation"

OK, so Einstein's derivation is lacking; I thought so. And you don't have to take my word for it.

Unlike Dr. Lorentz and unlike Professor Dr. Einstein, I think there is a preferred reference frame for the speed of light! The preferred reference frame is called gravity. The local gravity provides the aether for light.