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On Lorentz transforms.


Mitcher
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Lorentz transformations are important and needed only if the distances between objects are observed, which leads to the accounting of acceleration and energy. That is, we are talking about external observation in relation to objects. At the same time, the objects themselves are considered as material points.
But it has already became possible to consider objects as points of accumulation, that is all changes are considered through what is happening inside the object. Then the Lorentz transformations are no longer needed.
There are two theories in Analytic Philosophy, External and Internal Relations. The first works with external observations, the second with internal. 

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31 minutes ago, Ilya Geller said:

Lorentz transformations are important and needed only if the distances between objects are observed, which leads to the accounting of acceleration and energy. That is, we are talking about external observation in relation to objects. At the same time, the objects themselves are considered as material points.
But it has already became possible to consider objects as points of accumulation, that is all changes are considered through what is happening inside the object. Then the Lorentz transformations are no longer needed.
There are two theories in Analytic Philosophy, External and Internal Relations. The first works with external observations, the second with internal. 

Yes, my interest here was purely in terms of observed intervals of space-time and what it implies as a fundamental level. We definitely have a huge problem there since space and time, which are the bases of all our units, are circular defined. There are no intrinsic definitions for them.

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3 minutes ago, Mitcher said:

Yes, my interest here was purely in terms of observed intervals of space-time and what it implies as a fundamental level. We definitely have a huge problem there since space and time, which are the bases of all our units, are circular defined. There are no intrinsic definitions for them.

Exactly! Right to the point! This is the reason why I proposed my Quantitative theory — to count the number of elements in bodies and register external changes through changes within bodies, within the number. Indeed, the number of elements is a Natural number. Therefore, the measurement of quantity is objective! Then you can abandon the measurement of distance, speed and other things (like energy) because it is subjective, it is done in the term of Irrational numbers.

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  • 2 weeks later...
On 8/21/2022 at 12:03 PM, Mitcher said:

I'am still working on it and I'am not sure somebody tried to show Lorentz transforms, ds invariance and velocities composition in a single coherent numerical piece of work.

On 8/18/2022 at 3:45 PM, Mitcher said:

I don't conform that it's not possible to draw a simple working space-time diagram where Lorentz relationships, velocities addition and conserved ds would be obvious. I understand of course that you would need many more dimensions than a piece of paper can offer.

On 8/20/2022 at 11:39 AM, Mitcher said:

Hi. With the help of the many answers I got from people here and the numerical exemple from Peter Collier’s book I was able to close this simple calculation for the first time, founding v’ = - v and ds = ds’ as it should be. Before that I was mixing v, x and t the wrong way. Only glitch is that he is using c = 1 and hence c squared also equals 1 but I think it will work if one use any number larger than 1 and v.

Is this ds arclength?

Quote

es, my interest here was purely in terms of observed intervals of space-time and what it implies as a fundamental level. We definitely have a huge problem there since space and time, which are the bases of all our units, are circular defined. There are no intrinsic definitions for them.

You're speaking here under relativity's concern with spacetime bound together? Is that why you say they're circular defined?

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Sorry, this is basic Lorentz ds, I see from Studiot in a book by Eddington,
"The interval
svg.image?ds between two neighbouring events with coordinates svg.image?(x_{1},x_{2},x_{3},x_{4}) andsvg.image?(x_{1}+dx_{1},x_{2}+dx_{2},x_{3}+dx_{3},x_{4}+dx_{4}) in any coordinate system is given by svg.image?ds^{2}=g_{11}dx_{1}^{2}+g_{22}dx_{2}^2+g_{33}dx_{3}^2+g_{44}dx_{4}^{2}+2g_{12}dx_{1}dx_{2}+2g_{13}dx_{1}dx_{3}+2g_{14}dx_{1}dx_{4}+2g_{23}dx_{2}dx_{3}+2g_{24}dx_{2}dx_{4}+2g_{34}dx_{3}dx_{4}.......

Where the coefficients svg.image?g_{11}, etc. are functions of x1,x2,x3,x4. That is to say, dsis some quadratic function of the differences of coordinates.

Edited by NTuft
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9 hours ago, NTuft said:

Sorry, this is basic Lorentz ds, I see from Studiot in a book by Eddington,
"The interval
svg.image?ds between two neighbouring events with coordinates svg.image?(x_{1},x_{2},x_{3},x_{4}) andsvg.image?(x_{1}+dx_{1},x_{2}+dx_{2},x_{3}+dx_{3},x_{4}+dx_{4}) in any coordinate system is given by svg.image?ds^{2}=g_{11}dx_{1}^{2}+g_{22}dx_{2}^2+g_{33}dx_{3}^2+g_{44}dx_{4}^{2}+2g_{12}dx_{1}dx_{2}+2g_{13}dx_{1}dx_{3}+2g_{14}dx_{1}dx_{4}+2g_{23}dx_{2}dx_{3}+2g_{24}dx_{2}dx_{4}+2g_{34}dx_{3}dx_{4}.......

Where the coefficients svg.image?g_{11}, etc. are functions of x1,x2,x3,x4. That is to say, dsis some quadratic function of the differences of coordinates.

The quote from Eddington was a discussion of why the general quadratic is not used in relativity.

 

For those who want a modern accessible mathematical treatment I recommend MacComb.

Since I see that there has been some discussion about light cones and causality and relativity diagrams that are simple here is his version.

The book is great as it goes right the way through from Gaileo, Newton, Lorenz, Einstein, FourVectors, Simple GR geometry and curved triangulation, yet is rigerous enough.

Some fun relativistic calculations include the relativistic Compton Effect and "When Photon meets Proton head on".

McComb.jpg.fa3f74f32b2152cdae95a24d610eb984.jpgintervals_causality.jpg.76e4e044c9c3531264cebf51c8085e2d.jpg

 

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On 9/19/2022 at 12:32 AM, NTuft said:

Is this ds arclength?

I'am not using hyperbolic trig here.

On 9/19/2022 at 12:32 AM, NTuft said:

Is this ds arclength?

You're speaking here under relativity's concern with spacetime bound together? Is that why you say they're circular defined?

Not even that, I'am talking about the most basic definition : distance is the space interval covered in a certain time by a mobile with velocity v, and time is the duration measured by a clock of a mobile covering a certain distance with velocity v. This is the best we have to define space and time.

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On 9/19/2022 at 12:35 PM, studiot said:

Since I see that there has been some discussion about light cones and causality and relativity diagrams that are simple here is his version.

Note that in the diagram you posted above it is not specifically stipulated which path a photon emitted at the vertex will follow. Is it the space line X or the space-time x = ct one ? Not everybody agreed on this here.

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11 hours ago, Mitcher said:

time is the duration measured by a clock of a mobile covering a certain distance with velocity v. This is the best we have to define space and time.

What is the time measured by a clock moving 0 m with velocity 0 m/s? How would you define an hour measured with a stationary clock? If time can be measured without motion, why would the "most basic" definition require motion?

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4 hours ago, md65536 said:

What is the time measured by a clock moving 0 m with velocity 0 m/s? How would you define an hour measured with a stationary clock? If time can be measured without motion, why would the "most basic" definition require motion?

If there is no motion then you only have time to characterize time : time is the time you measure time is the time.... Same for space. With Mass they form the basis of all our dimensional units but we have no precise idea of what space and time intrinsic nature are. Once this is acknowleged then we should admit they could be fundamentally something very different. Is time really a 1D something binded onto a 4D something ? Is it linear, circular, limitless ? This is somehow a childish representation and physicists are starting to admit it, however who is working on researching the true nature of space-time ?

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On 9/21/2022 at 10:12 AM, md65536 said:

What is the time measured by a clock moving 0 m with velocity 0 m/s? How would you define an hour measured with a stationary clock? If time can be measured without motion, why would the "most basic" definition require motion?

On 9/21/2022 at 3:04 PM, Mitcher said:

If there is no motion then you only have time to characterize time : time is the time you measure time is the time.... Same for space. With Mass they form the basis of all our dimensional units but we have no precise idea of what space and time intrinsic nature are. Once this is acknowleged then we should admit they could be fundamentally something very different. Is time really a 1D something binded onto a 4D something ? Is it linear, circular, limitless ? This is somehow a childish representation and physicists are starting to admit it, however who is working on researching the true nature of space-time ?

to paraphrase @swansont, "time is the integral of frequency".

Quote

 

For cyclical processes, such as rotation, oscillations, or waves, frequency is defined as a number of cycles per unit time. In physics and engineering disciplines, such as optics, acoustics, and radio, frequency is usually denoted by a Latin letter f or by the Greek letter svg.image?\nu or ν (nu) (see e.g. Planck's formula).

The relation between the frequency and the period,T, of a repeating event or oscillation is given by
svg.image?f=\frac{1}{T}

 

 f, units Hz, or s(econds)-1, ergo, period T in units s1

I think vibration is more fundamental than distance. What about distance as in wavelength?

I don't think I agree that time needs a motion -- it sounds like you're referencing speed. I think I agree with @md65536 that we can simply, arbitrarlly, define a period, and then we have a time period... though in some fashion this may entail an abstract clock.

Now, as for space, @Mitcher, do you suppose we can look at geometry à la mode de group theory, as non-circular defined? Could it not be said it is self-sufficient, that it is defined through mathematics, or do you suppose there is recursion there to circular definition?

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2 hours ago, NTuft said:

to paraphrase @swansont, "time is the integral of frequency".

 f, units Hz, or s(econds)-1, ergo, period T in units s1

I think vibration is more fundamental than distance. What about distance as in wavelength?

I don't think I agree that time needs a motion -- it sounds like you're referencing speed. I think I agree with @md65536 that we can simply, arbitrarlly, define a period, and then we have a time period... though in some fashion this may entail an abstract clock.

Now, as for space, @Mitcher, do you suppose we can look at geometry à la mode de group theory, as non-circular defined? Could it not be said it is self-sufficient, that it is defined through mathematics, or do you suppose there is recursion there to circular definition?

The integral, with respect to t,  of 1/t is: ln t +C, surely?  

Edited by exchemist
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On 9/21/2022 at 6:04 PM, Mitcher said:

If there is no motion then you only have time to characterize time : time is the time you measure time is the time

And yet we don't rely on motion to characterize time; great effort has been made to reduce motion of atoms or ions in clocks over the years, since it contributes to a frequency error in the clocks.

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On 9/20/2022 at 2:14 PM, Mitcher said:

I'am not using hyperbolic trig here.

[...]

I see you say, "...the classical Lorentz...". To reformulate, what is this ds; ds invariance you refer to?

6 hours ago, exchemist said:

The integral, with respect to t,  of 1/t is: ln t +C, surely?  

I think we have to ask, are we actually looking at a function there? We have an inverse relation equated, and I proposed that we were going to select a value, T, to create an arbitrary frequency or 'time period'. I don't think the bare calculus does us much good here, but I always think it is worth the exercise. 

What is it we would want to see from this integral? I'd say we want the summation of successive periods, creating an interval of time. Similarly, if we had it graphed appropriately, we could say we want the area under the curve. However, graphing the function here as 1/x (and note: our T selected above is not a variable, per se, we had chosen a specific value) we have I think the rectangular hyperbola. I do not think this function serves us for our abstract idea of a time period, but maybe I am not thinking broadly enough.

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8 hours ago, swansont said:

And yet we don't rely on motion to characterize time; great effort has been made to reduce motion of atoms or ions in clocks over the years, since it contributes to a frequency error in the clocks.

You are speaking about measuring intervals of time. Time and energy and space lack comprehensive definitions that would include their fundamental nature, yet we can measure them with smaller and smaller accuracy so they don't seem to form a lattice albeit it cannot be excluded either. So what is it ?

14 hours ago, NTuft said:

think vibration is more fundamental than distance. What about distance as in wavelength?

I don't think I agree that time needs a motion -- it sounds like you're referencing speed. 

Vibration is precisely a form of motion, and yes, I definitely reference to speed. A time-measuring device has to be in motion, it merely compares motions between them and gives the measurement in ratios. The clock turns in phase with Earth for instance.

On 9/21/2022 at 7:12 PM, md65536 said:

What is the time measured by a clock moving 0 m with velocity 0 m/s?

If a clock is not moving then it is broken and cannot measure anything. This would apply for an atomic clock or any other technologies.

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4 hours ago, Mitcher said:

You are speaking about measuring intervals of time. Time and energy and space lack comprehensive definitions that would include their fundamental nature, yet we can measure them with smaller and smaller accuracy so they don't seem to form a lattice albeit it cannot be excluded either. So what is it ?

I can’t parse this. What does this have to do with the discussion?

4 hours ago, Mitcher said:

Vibration is precisely a form of motion, and yes, I definitely reference to speed. A time-measuring device has to be in motion, it merely compares motions between them and gives the measurement in ratios. The clock turns in phase with Earth for instance.

If a clock is not moving then it is broken and cannot measure anything. This would apply for an atomic clock or any other technologies.

Motion isn’t absolute, so what motion is required? 

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