# Pythagoras's Theorem ideal

Dappy
Pythagoras's Theorum is considered to be an ideal. The accuracy of its results are dependent upon the flatness of the surface to which it is being applied. Since there is no such thing as a perfectly flat surface, Pythagoras's Theorum results can only be seen as an approximation. Is this true in its application to Lorentz transformation?

## Answers and Replies

Mentor
The theorem is exact, you just have to keep its requirements in mind. Triangles in the real world are just approximations to perfect [strike]rectangular[/strike] right triangles, but that is not part of the theorem. Real triangles are not three ideal lines connected at three ideal points anyway.

In special relativity, space is flat. The theorem can be applied by every observer, if the (idealized) triangle has a right angle for this observer (that is frame-dependent).

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Dappy
-In special relativity, space is flat. The theorem can be applied by every observer, if the (idealized) triangle has a right angle for this observer (that is frame-dependent).

In General relativity space time is curved. I'm confused. Surely the flat space of SR must also be an ideal.

Mentor
In General relativity space time is curved. I'm confused. Surely the flat space of SR must also be an ideal.
Sure, special relativity is general relativity without gravity. You asked about Lorentz transformations, they are mainly a tool of special relativity.

Staff Emeritus
Homework Helper
I'm still working on what a 'rectangular triangle' is.

Mentor
Since there is no such thing as a perfectly flat surface, Pythagoras's Theorem results can only be seen as an approximation.
It's the other way around - Pythagoras's theorem is exact and real triangles drawn on real surfaces in the real world are approximations of that ideal.

This will be true of just about any application of mathematics to engineering and experimental science.

Dappy
It's the other way around - Pythagoras's theorem is exact and real triangles drawn on real surfaces in the real world are approximations of that ideal.

This will be true of just about any application of mathematics to engineering and experimental science.

Pythagoras's Theorum is exact and therefore can only give an approximate
result to a real situation in an imperfect world. I think we're saying the same thing. I'm confused again.

Mentor
I'm still working on what a 'rectangular triangle' is.
I'm still working on my second language.

Dappy said:
Pythagoras's Theorum is exact and therefore can only give an approximate
result to a real situation in an imperfect world.
There is no "therefore". Nothing can give exact results in the real world, as there are no exact values or measurements.

dauto
Would you please stop saying Theorum? It's really distracting. Why not say theorem?

Dappy
There is no "therefore". Nothing can give exact results in the real world, as there are no exact values or measurements.

Can we trust Lorentz transformations as being accurate? Pardon me for asking, but what is your first language?

Dappy
Would you please stop saying Theorum? It's really distracting. Why not say theorem?

My apologies, theorem.

Would you please stop saying Theorum? It's really distracting. Why not say theorem?

What's distracting are your irrelevant and pointless comments.

Mentor
2021 Award
This thread is drifting into philosophy. If you wish to discuss "perfection" here then you need to start with a scientific definition of perfection.

dauto
What's distracting are your irrelevant and pointless comments.

I didn't make any comments. I asked Dappy for a favor and he was happy to oblige. Thank you Dappy. Now who is making pointless comments.

Dappy
This thread is drifting into philosophy. If you wish to discuss "perfection" here then you need to start with a scientific definition of perfection.

Thankyou

We, I assume, accept that Pythagoras's theorem is an ideal. Is that ideal realized in the real world when calculating Lorentz transformations for time, length and relativistic mass?

Mentor
2021 Award
Provided gravitational effects are not significant, then yes.

Mentor
Can we trust Lorentz transformations as being accurate?
If you take gravitational effects into account, no experiment has ever found a deviation from Lorentz transformations. And there are experiments testing that with a precision of better than one part in a billion.

Pardon me for asking, but what is your first language?
German.

Dappy
If you take gravitational effects into account, no experiment has ever found a deviation from Lorentz transformations. And there are experiments testing that with a precision of better than one part in a billion.

That's almost good enough for me. Can you imagine how flat a surface would have to be for Pythagoras's theorem to be accurate to better than one part in a billion. The mirror in the Hubble telescope comes to mind. I wonder how accurate it is in that context.

I don't know how to double quote, but German, I would never of guest. I'm dyslexic and slip up a lot more than that.

Reagards Dappy.

Mentor
Can you imagine how flat a surface would have to be for Pythagoras's theorem to be accurate to better than one part in a billion.
Flatter than one part in a billion? :D
To have length deviations of one part in a billion on a perfect sphere with the size of earth, the triangle needs an area of ~0.05 km^2.

1 part in a billion is just a random number, the quality of the tests depends on the tested quantity. The stability of the speed of light in different directions can be tested way better (wikipedia gives 1 part in 1017).
The next generation of ring lasers is expected to measure the rotation of earth more precise than satellites and telescopes. That's a 1 part in 1 billion measurement for an already tiny effect.

More: Modern searches for Lorentz violation

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Mentor
On a curved 2D surface, the Pythagorean Theorem cannot be accurately applied over large distances. However, over short distances, it can provide an excellent approximation. Thus, on the surface of the earth, it will not be accurate over large variations of latitude and longitude. However, over short distances like a few miles, it typically provides an excellent approximation. In space-time, the analog of the Pythagorean theorem is the Minkowski metric. Over large spans of space-time in a region of large mass-induced curvature, it is not possible to define a set of coordinates that accurately describes distances using the Minkowski metric. However, over small local regions, it is possible.

Dappy
Flatter than one part in a billion? :D

That isn't quite what I said, but it made me laugh and when I realized what I actually said, it was equally ridiculous.
Now that you've pointed that out, I have to say "right angled triangle". ;-)

Thankyou mfb for the link. I found it most informative and need to give it more attention, to allow it all to soak in.

Dappy
Over large spans of space-time in a region of large mass-induced curvature, it is not possible to define a set of coordinates that accurately describes distances using the Minkowski metric.

So can I assume that Lorentz violation may exist there, but only because of our inability to define a set of coordinates?

Mentor
That is not considered as a violation of Lorentz invariance - it is just a setup where Lorentz transformations are not the right tool for the whole space. You need general relativity then.

Dappy
I think I understand now. Lorentz transformation is the application of Pythagoras's theorem to two distance ratios, 1:v/c where 1 is the speed of light. The fact that the speed of light is invariant means that we can trust that 1 will always be 1 and hence Lorentz invariant.