# Can this object be moved using a Lorentz transform?

#### steveupson

There's a 3D object which has been created that contains certain features, the main feature of which is a defined relationship between some defined planes.

#### topsquark

Math Team
There's a 3D object which has been created that contains certain features, the main feature of which is a defined relationship between some defined planes.

The link seems to have moved on, though I did appreciate the "Pirate Banana Song." :dance:

As a Lorentz transformation doesn't move an object, just the reference frame it is observing it from, then what are you talking about? I can't think of any object that would be defined such that a Lorentz transformation would "not preserve it" in whatever manner.

-Dan

#### steveupson

I can't think of any object that would be defined such that a Lorentz transformation would "not preserve it" in whatever manner.

-Dan
Of course not, and that's exactly why I'm asking. The video is only 4 seconds and is a fragment of the animation that we are currently working on. It works when I try it.

edited to add> A few months ago I did not imagine that there existed such an object, but now I'm pretty sure that it does exist.

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#### topsquark

Math Team
Okay, I understand about the video now. But the graphic still doesn't tell me what's going on?

-Dan

#### steveupson

Yes, that's the difficulty that I'm having. How can I possibly explain what I'm talking about? Thanks for your patience.

The object represents a function that exists between two angles, each of which is formed between two planes. It's a novel approach. Nothing like it exists in any literature that's out there.

If I can walk you through exactly how the object is created, then we might be able to move on to discuss what makes the object new and unique, and why I think it would resist all attempts at transformation.

The first angle that we are concerned with is the elevation angle E. It is formed when we intersect a sphere with two planes, like this:

https://youtu.be/ho8XCHIT-Oo

The second angle is more complicated. It is formed between a plane of longitude and a "tangent" plane which lies in a conical orbit. The construction of the longitude plane is not too complicated, but its orientation is very specific. The orientation of the longitude plane is such that it intersects a point on a small circle which is constructed on the surface of the sphere at a 45 degree angle, like this:

https://youtu.be/xwjIeHC3Nb0

The "tangent" plane lies along the surface of a cone formed by the sphere center and the 45 degree small circle. As the elevation angle E is varied, the position of this tangent plane changes such that it remains coincident with the intersecting point on the circumference of the small circle, like this:

https://youtu.be/6OnSZki9yp0[/U]

If we call the angle that is made between the longitude plane and the tangent plane the angle $$\displaystyle \alpha$$, then the object is defined by the function:

$$\displaystyle \cot( \alpha) = \dfrac{1-\sin(E)}{\sin(E)}$$

At least I think it is. My algebra is pathetic.

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#### topsquark

Math Team
Okay, just to check: Your object can only be composed by certain sets of defined angles, at least some of which will deform under a Lorentz transformation and thus no longer be definable as that type of object?

Sounds more like a symmetry problem than a transformation problem.

-Dan

#### steveupson

You're probably correct, but it doesn't sound quite right to me. The way that I understand it (or not), an equation would be true in any frame, regardless of boost. In this case, the equation under transformation has only one quantity, and that quantity expresses the property of direction. There is no metric to it, if I understand the meanings correctly. It's a 3-dimensional object expressed in non-dimensional units. At least that's how I would categorize it.

Since it has no metric, it wouldn't deform, would it? My question is whether or not the transformation would produce the same direction information as the original. Wouldn't the transformation give an alternative set that is parallel to the original, ie. not the same?

#### topsquark

Math Team
You're probably correct, but it doesn't sound quite right to me. The way that I understand it (or not), an equation would be true in any frame, regardless of boost. In this case, the equation under transformation has only one quantity, and that quantity expresses the property of direction. There is no metric to it, if I understand the meanings correctly. It's a 3-dimensional object expressed in non-dimensional units. At least that's how I would categorize it.

Since it has no metric, it wouldn't deform, would it? My question is whether or not the transformation would produce the same direction information as the original. Wouldn't the transformation give an alternative set that is parallel to the original, ie. not the same?
In SR all reference frames have the same metric. I'm still not sure what the issue here is. A sphere in a static coordinate system will look like some kind of ellipse when viewed by a moving inertial reference frame. I was wondering if that kind of "deformation" was giving a problem. If not then I'm not understanding what your difficulty is. Sorry, my brain is numb tonight.

-Dan

#### steveupson

In SR all reference frames have the same metric. I'm still not sure what the issue here is. A sphere in a static coordinate system will look like some kind of ellipse when viewed by a moving inertial reference frame. I was wondering if that kind of "deformation" was giving a problem. If not then I'm not understanding what your difficulty is. Sorry, my brain is numb tonight.

-Dan
The thing is, there is no sphere.

There is no two-dimensional surface, other than planes.

And a circle on a plane, but no spheres.