# Non-standard clock synchronization

• bernhard.rothenstein
In summary: Just amazing.In summary, Selleri's transformation equations propose that x'=g(x'-vt') and t=t'/g, where g=gamma. Einstein's clock synchronization requires t(E) to be equal to t(e)+x/c, which can be achieved by reading a clock located at a distance x from the origin when the synchronizing light signal is emitted.

#### bernhard.rothenstein

Selleri proposes the following transformation equations
x=g(x'-vt')
t=t'/g
g=gamma.
Einstein's clock synchronization requires
t(E)=t(e)+x/c
t(E) reading of a clock located at a distance x from the origin, t(e) reading of a clock located at the origin when the synchronizing light signal is emitted,
Is there a link between the proposed transformation equations and the synchronization procedure?

bernhard.rothenstein said:
Selleri proposes the following transformation equations
x=g(x'-vt')
t=t'/g
g=gamma.
Einstein's clock synchronization requires
t(E)=t(e)+x/c
t(E) reading of a clock located at a distance x from the origin, t(e) reading of a clock located at the origin when the synchronizing light signal is emitted,
Is there a link between the proposed transformation equations and the synchronization procedure?

DrGreg has already answered this question in a very beautiful way here.

nonstandard clock synchronization

1effect said:
DrGreg has already answered this question in a very beautiful way here.
Thanks. I appreciated DrGreg's answer. But what I mention are not Tangherlini but Selleri transformasions.

bernhard.rothenstein said:
Thanks. I appreciated DrGreg's answer. But what I mention are not Tangherlini but Selleri transformasions.

They are one and the same thing.

1effect said:
They are one and the same thing.
Yes (except that the roles of "primed" and "unprimed" have been reversed). "Tangherlini Transform", "Selleri Transform", "Generalised Galilean Transform" (GGT), all refer to the same thing (though different authors attribute different significances to them). They also sometimes appear under the heading of "Lorentz Ether Theory" (LET), although Lorentz never used them.

Just to spell out the post that 1effect referred to:

Given Einstein coords (x, t) for observer I, define Tangherlini coords (x'T, t'T) for observer I'T by

x'T = g(x - vt)
t'T = t/g
g = gamma = (1 - v2/c2)-1/2

Then use the Lorentz transform (and its inverse) for Einstein-observer I'E (x'E, t'E)

x'E = g(x - vt)
t = g(t'E + vx'E/c2)

and you will obtain

x'T = x'E
t'T = t'E + vx'E/c2

Note that c must be interpreted as the speed of light in the I frame, or equivalently the "two-way speed of light" in the I'T frame, because, in that frame the "one-way speed of light" depends on which direction you measure it in.

DrGreg said:
Yes (except that the roles of "primed" and "unprimed" have been reversed). "Tangherlini Transform", "Selleri Transform", "Generalised Galilean Transform" (GGT), all refer to the same thing (though different authors attribute different significances to them). They also sometimes appear under the heading of "Lorentz Ether Theory" (LET), although Lorentz never used them.

Note that c must be interpreted as the speed of light in the I frame, or equivalently the "two-way speed of light" in the I'T frame, because, in that frame the "one-way speed of light" depends on which direction you measure it in.

Thank you, DrGreg

I am not familiar with the "one-way speed" dependency on the direction of measurement. Does this follow from the definition of the metric in the Tangherlini/Selleri theory? Can you elaborate?

1effect said:
I am not familiar with the "one-way speed" dependency on the direction of measurement. Does this follow from the definition of the metric in the Tangherlini/Selleri theory? Can you elaborate?
Consider the equation x'E = ct'E. That describes something moving at the speed of light measured in the I'E frame. Now convert to the I'T frame. You get

x'T = ct'E

t'T = t'E + vt'E/c = (1 + v/c)t'E

From which the Tangherlini-speed of light is c / (1 + v/c).

Similarly, using x'E = -ct'E, the speed in the other direction is c / (1 - v/c).

But the there-and-back (2-way) average speed of light is still c.

You could also deduce this from the metric I quoted here, by solving ds = 0.

In fact, whenever the metric contains "off-diagonal" terms which "mix space and time", like dxdt, the speed of light, expressed in those coordinates, cannot be isotropic (=the same in all directions). (Think about solving ds = 0 and the roots of quadratic equations.)

DrGreg said:
Consider the equation x'E = ct'E. That describes something moving at the speed of light measured in the I'E frame. Now convert to the I'T frame. You get

x'T = ct'E

t'T = t'E + vt'E/c = (1 + v/c)t'E

From which the Tangherlini-speed of light is c / (1 + v/c).

Similarly, using x'E = -ct'E, the speed in the other direction is c / (1 - v/c).

But the there-and-back (2-way) average speed of light is still c.

You could also deduce this from the metric I quoted here, by solving ds = 0.

In fact, whenever the metric contains "off-diagonal" terms which "mix space and time", like dxdt, the speed of light, expressed in those coordinates, cannot be isotropic (=the same in all directions). (Think about solving ds = 0 and the roots of quadratic equations.)

Thank you, this is what I thought :

$$x=g(x'-vt')$$
$$t=t'/g$$

$$0=x^2-(ct)^2=g^2(x'-vt')^2-(ct'/g)^2$$

Solving
$$g^2(x'-vt')^2-(ct'/g)^2=0$$ for $$\frac{x'}{t'}$$ I get the anysotropic speed.

You are right, the presence of the mixed term in x't' results immediately in anysotropic light speed.
Thank you so much for the incredible stuff, you are awesome!

bernhard.rothenstein said:
Selleri proposes the following transformation equations
x=g(x'-vt')
t=t'/g
g=gamma.
Einstein's clock synchronization requires
t(E)=t(e)+x/c
t(E) reading of a clock located at a distance x from the origin, t(e) reading of a clock located at the origin when the synchronizing light signal is emitted,
Is there a link between the proposed transformation equations and the synchronization procedure?
Sorry, I just realized I never gave an explicit answer to this question.

The answer follows from my formula for the "one-way Selleri-speed" of light c / (1 +/- v/c).

So in your notation

$$t'_{T}(E)=t'_{T}(e)+\left|x\right|\frac{1+\frac{v}{c}}{c}$$

where c is the "two-way speed of light".

DrGreg said:
Sorry, I just realized I never gave an explicit answer to this question.

The answer follows from my formula for the "one-way Selleri-speed" of light c / (1 +/- v/c).

So in your notation

$$t'_{T}(E)=t'_{T}(e)+\left|x\right|\frac{1+\frac{v}{c}}{c}$$

where c is the "two-way speed of light".

Please give me some hints for deriving by myself the equation you propose.

bernhard.rothenstein said:
Please give me some hints for deriving by myself the equation you propose.
It follows from several previous posts.

ONE

This post showed that the 1-way Selleri-speed of light is

c+ = c / (1 + v/c)

(where c is the 2-way speed and also the 1-way Einstein-speed).

TWO

From this post I thought you already knew that

t(E) = t(e) + x/c+ ... (1)

However, I've just noticed your descriptions of t(E) and t(e) in that post are confusing and maybe not quite what I am assuming. To avoid all doubt, I am interpreting:

t(e) to be the time, on the observer's own clock, when a light signal is emitted by the observer at the spatial origin.

t(E) to be the time, on an auxiliary clock located at a constant distance x from the observer, when the light signal is received.

The auxiliary clock is synchronised to make the equation (1) true.

For a given choice of c+ (between 0.5 and infinity), the set of all possible auxiliary clocks defines a (t,x) coordinate system.

Conclusion

Put the above two posts together to get my result.

The reason I originally put |x| instead of x is because of a technicality I haven't raised yet.

If you define your synchronisation operationally via light from the observer, then equation (1) would hold only for positive x. For negative x you'd need to replace x by -x. That's assuming that c+ is the 1-way speed of light outward away from the observer.

However, you might instead want c+ to be the 1-way speed of light in the positive x direction. In this case, equation (1) is correct even for negative x, but then the light is really traveling from the auxiliary clock to the observer (still in the positive x direction). In the case of Selleri coords, this second interpretation is the correct one, so I was wrong to use |x| in my previous post.

From a practical point of view, for negative x you can use the alternative equation

t(E) = t(e) - x/c- ... (2)

where c- is the 1-way speed of light in the negative x direction.

c+ and c- are related by 1/c = (1/c+ + 1/c-)/2 (see this post).

## 1. What is non-standard clock synchronization?

Non-standard clock synchronization is a method used to synchronize clocks in a system that does not follow the traditional method of using a central clock or time server. Instead, each clock in the system is synchronized with one or more other clocks in the system, creating a decentralized synchronization network.

## 2. How does non-standard clock synchronization work?

In non-standard clock synchronization, each clock in the system sends and receives synchronization signals to and from other clocks in the system. These signals are used to adjust the time on each clock, ensuring that they are all in sync with each other.

## 3. What are the advantages of non-standard clock synchronization?

Non-standard clock synchronization offers several advantages over traditional methods of clock synchronization. It allows for a more flexible and adaptable system, as well as increased accuracy and precision in timekeeping. It is also less vulnerable to failures of a central time server.

## 4. What are the potential drawbacks of non-standard clock synchronization?

Non-standard clock synchronization can be more complex to implement and maintain compared to traditional methods. It also requires a reliable network and can be impacted by network disruptions or delays. Additionally, there may be a lack of standardization in non-standard clock synchronization methods.

## 5. How is non-standard clock synchronization used in practice?

Non-standard clock synchronization is commonly used in distributed systems, such as computer networks, to ensure that all devices have accurate and synchronized time. It is also used in scientific research, such as in experiments that require precise timing or data logging across multiple locations.