Is Euclid's 5th Postulate Correct? - A Teacher's Perspective

[AGuglielmone]

Let me begin by saying that I am a special ed teacher (moderate to severe disabilities) and don’t have much in the way of a maths/physics background.

If mass curves spacetime and the other fundamental forces create similar fields is it the case that in the real world Euclid’s 5th postulate is actually correct? What I mean to say is that an orbit is actually straight line parallel motion but is perceived as circular because we cannot see the distortion caused by the field.

If all orbits are actually parallel linear translation across time then is it not the case that orbiting bodies do not experience any acceleration? Are orbits from electrons to those of planets just two bodies that experience time change at the same rate?

I hope this makes some sort of sense If not ignore me lol.

Cheers,

Tony

 

[Peter Morgan]

For the Euclid's 5th postulate part of your post, we first have to note that Euclidean geometry is about a 3-dimensional space, without time, and certainly not about Minkowski space. That said, I think we'd have to say that Euclid's 5th postulate doesn't hold in a curved space-time, insofar as two light rays starting from a very distant star (that is, they are two intersecting lines) and passing on opposite sides of a massive object may intersect again. We actually see such a situation when we observe two copies of a distant star or galaxy.

Bodies that are not subject to electromagnetic, electroweak, strong, or other forces do not experience acceleration. Which is the content of the empirically very well established equivalence principle. A rapidly varying gravitational field may cause a body to experience shear forces, however.

I regret that I don't understand what you mean by the last question.

 

[kent davidge]

I think the OP asks whether or not Euclid's 5th postulate certainly holds in any place in space (yes, any place in space, because as Peter Morgan notes, the 5th postulate has as background our three-dimensional space). The answer is, no. As mass/energy curves space-time, it generally will curve space, which means the 5th postulate will not hold. A great example of this is the deflection by the sun of star light, which by the way was proposed by Einstein himself as a way to test his theory.

 

[PeroK]

Let me begin by saying that I am a special ed teacher (moderate to severe disabilities) and don’t have much in the way of a maths/physics background.

If mass curves spacetime and the other fundamental forces create similar fields is it the case that in the real world Euclid’s 5th postulate is actually correct? What I mean to say is that an orbit is actually straight line parallel motion but is perceived as circular because we cannot see the distortion caused by the field.

If all orbits are actually parallel linear translation across time then is it not the case that orbiting bodies do not experience any acceleration? Are orbits from electrons to those of planets just two bodies that experience time change at the same rate?

I hope this makes some sort of sense If not ignore me lol.

Cheers,

Tony

Euclid's postulates relate to plane geometry, a branch of axiomatic mathematics. The 5th postulate cannot be deduced from the other four postulates. It is neither right nor wrong. It is an axiom that, when accepted, leads to Euclidean geometry. Euclid's geometry is independent of the nature of space in our universe.

The only relevant question is whether space is Euclidean or not. It is not. But, that has no mathematical impact on Euclidean geometry.

 

[Nugatory]

If mass curves spacetime and the other fundamental forces create similar fields is it the case that in the real world Euclid’s 5th postulate is actually correct?

There are two problems with this line of thought.

First, despite the superficial similarity of Newton's inverse-square law for gravitation and Coulomb's inverse-square law for electrostatics, gravity is the only force that can be explained by spacetime curvature. We have a few older threads on this, but the basic idea is this: At any given point in spacetime there is only "straight line" in any given direction, so curvature effects must affect all objects in the same way regardless of their internal composition. That's true of gravity - Galileo's famous experiment showing that all dropped objects fall at the same speed is an example - but egregiously untrue for electrical and magnetic forces which vary with the charge and speed of an object.

A second problem is that the "straight lines" of general relativity (which are properly called "geodesics") don't necessarily obey Euclid's fifth postulate. Look at the diagram at the top of the wikipedia article on the fifth postulate; if you understand these as geodesics in spacetime the two lines do not necessarily have to meet. In other words, and as @PeroK has said above, that tells us that spacetime is not Euclidean.

If all orbits are actually parallel linear translation across time then is it not the case that orbiting bodies do not experience any acceleration? Are orbits from electrons to those of planets just two bodies that experience time change at the same rate?

Electrons do not orbit in any way, shape, or fashion. The idea that they do is so deeply engrained in the popular imagination that it even appears in our forum logo - but in fact that picture was disproven almost a century ago. Google for "planetary atomic model" for more information.

 

[sweet springs]

Euclid 5th postulate makes Euclid geometry. Real physics adopts Rieman geometry that do no apply this postulate.

 

[pervect]

Let me begin by saying that I am a special ed teacher (moderate to severe disabilities) and don’t have much in the way of a maths/physics background.

If mass curves spacetime and the other fundamental forces create similar fields is it the case that in the real world Euclid’s 5th postulate is actually correct? What I mean to say is that an orbit is actually straight line parallel motion but is perceived as circular because we cannot see the distortion caused by the field.

If all orbits are actually parallel linear translation across time then is it not the case that orbiting bodies do not experience any acceleration? Are orbits from electrons to those of planets just two bodies that experience time change at the same rate?

I hope this makes some sort of sense If not ignore me lol.

Cheers,

Tony

You ask about orbits. An orbit is essentially the equivalent of a straight line, called a geodesic, in a curved space-time.

You ask about the parallel postulate in the "real world". If we look at what our current best theory of gravity, General Relativity, says about the geometry of space near a massive object, it says that in general the geometry of space is not Euclidean. (There can be some exceptions where it can be, but in general it is not). Note that I'm talking about the geometry of space, and not space-time. It gets a bit technical to describe exactly what I mean by the geometry of space. Different readers (especially with some technical background) might wonder exactly how one moves from the geometry of space-time to the geometry of space. Basically, when I talk about the geometry of space, I am considering the easiest situation, where the geometry of space-time is independent of time, in particular the Schwarzschild geometry near a single, spherically symmetrical, non-rotating object. Then there is a meaningful notion of "at rest" in the space-time geometry, and there is a notion of space described by considering the distance relationships between points or objects "at rest" in the space-time geometry. This is what I mean by spatial geometry. Rather than taking the hard, general, route, I am talking about a specific example, to try to communicate some easy insights.

The spatial geometry, defined in this way, isn't Euclidean. It is, however, s a bit oversimplified to lay the blame of the failure of the geometry of space to be Euclidean entirely on Euclid's 5th postulate, the parallel postulate alone. If we relax only Euclid's parallel postulate, and keep the others, we do not get the sort of geometry we actually use in GR, which is Riemannian geometry.

If we stick to 2 dimensional geometry, and relax only the parallel postulate, we can talk about the geometry of a sphere, a plane, or a hyperbolic surface. But we can't yet talk about the geometry of the surface of a football. To do that, we need further modifications to the structure of Euclidean geometry beyond eliminating the parallel postulate.

Riemannian geometry turns out to be the sort of geometry that we need to handle the football, and is the sort of geometry that GR is built on.

What does this have to do with the real world? That gets a bit complicated. If we had massless, elastic strings, that were totally unaffected by gravity, those strings could draw "straight lines" in space for us in the static Schwarzschild geometry.

But - we don't have such idealized strings. So they are a bit of an abstraction. Gravity affects everything, the things we can actually build are affected by gravity. So we really study the real world by the behavior of light rays, and not by drawing straight lines. There have been several tests of how gravity affects light rays, including the original bending of light rays by the sun, and more sophisticated tests involving radar time delays (the Shaipiro effect). All these tests are compatible with GR, and with the notion that the geometry of space is curved. But there is a layer of interpretation here between the experiments we can carry out in the real world, and our abstract entities of straight lines and spatial geodesics.

 

[PeterDonis]

If we relax only Euclid's parallel postulate, and keep the others, we do not get the sort of geometry we actually use in GR, which is Riemannian geometry.

Actually, you do get Riemannian geometry in its strict interpretation (i.e., with a positive definite metric) if you relax the parallel postulate and keep the others. What you don't get is geometry with an indefinite metric, i.e., with timelike and null vectors as well as spacelike vectors. (This sort of geometry is more precisely termed "pseudo-Riemannian", although the term "Riemannian" is sometimes used even though it's not strictly correct.) That requires more than just relaxing the parallel postulate.