How Does a Spherical Earth Challenge Our Perception of Space and Geometry?

[Hyperspace2]

If there weren't hills,mountains etc, (that means purely spherical earth).
And suppose we weren't told Earth was round.

When we walk , we would think we are walking in never ending football ground(infinitely large playground ). Isn't this the effect of so called general relativity effect. (curved spacetime)

 

[ghwellsjr]

You would still know that the surface wasn't flat because the corners of your very large football field wouldn't all be 90 degrees. In fact, you could have a very large equilateral triangle with all three angles equal to 90 degrees instead of 60 degrees.

 

[Hyperspace2]

You would still know that the surface wasn't flat because the corners of your very large football field wouldn't all be 90 degrees. In fact, you could have a very large equilateral triangle with all three angles equal to 90 degrees instead of 60 degrees.

How a equilateral?
I guess I would see like I am in the middle of the the big football ground , and if I run for horizon I wouldn't meet them. I suppose I would feel like a rectangular planet ,isn't it?

 

[Dale]

How a equilateral?

Start on the equator, go due north to the north pole, turn 90º, go due south to the equator, turn 90º, go along the equator back to your starting point. The triangle thus formed has 3 equal-length sides and 3 angles of 90º each.

 

[HallsofIvy]

Draw a "line of longitude" from the north pole of your sphere to the equator at longitude, say, 0. Draw another line of longitude from the north pole to the equator at longitude 90 E. Each of those three lines will be 1/4 of the circumference of your spherical world and so will form an equilateral triangle, having three angles each of 90 degrees. In fact, on a sphere, any triangle has angles that add to more than 180 degrees.

Locally, as long as you don't go too far, any surface "feels" like a rectangle. That has nothing to do with relativity- people who used to believe the Earth was flat argued that way. Non-locally, if you keep walking in a straight line, you will eventually get back where you started- which is NOT true on a "rectangular planet".

 

[ghwellsjr]

Actually, it doesn't matter where you start or which direction you head in, if you walk 1/4 the circumference of the earth, make a 90 degree turn (either left or right), walk the same distance again and make make another identical turn of 90 degrees, and walk the same distance a third time, you will end up where you started, approaching at 90 degrees from your initial starting path. But these other explanations are perhaps easier to visualize.

But no one has answered the initial question, "Isn't this the effect of so called general relativity effect. (curved spacetime)"?

 

[Dale]

Mathematically it is related. The differences are:
1) the surface described would be a curved space embedded in a higher-dimensional flat space whereas there is no embedding space considered or required in GR
2) the surface described would be Riemannian, but curved spacetime is pseudo-Riemannian
3) the surface described is 2 dimensional whereas it is 1+3 dimensional for GR

 

[Hyperspace2]

You would still know that the surface wasn't flat because the corners of your very large football field wouldn't all be 90 degrees. In fact, you could have a very large equilateral triangle with all three angles equal to 90 degrees instead of 60 degrees.

Start on the equator, go due north to the north pole, turn 90º, go due south to the equator, turn 90º, go along the equator back to your starting point. The triangle thus formed has 3 equal-length sides and 3 angles of 90º each.

Draw a "line of longitude" from the north pole of your sphere to the equator at longitude, say, 0. Draw another line of longitude from the north pole to the equator at longitude 90 E. Each of those three lines will be 1/4 of the circumference of your spherical world and so will form an equilateral triangle, having three angles each of 90 degrees. In fact, on a sphere, any triangle has angles that add to more than 180 degrees.

Locally, as long as you don't go too far, any surface "feels" like a rectangle. That has nothing to do with relativity- people who used to believe the Earth was flat argued that way. Non-locally, if you keep walking in a straight line, you will eventually get back where you started- which is NOT true on a "rectangular planet".

Oh sorry do we see a circular or rectangular ground (I am little confused now)
Thanks for reply
Let me conclude
The rectangular ground(or circular ground) we consider would be actually a curved incomplete equilateral triangle .

You people mean that the locus of the horizon that observer see actually end up being curved equilateral triangle (as seen from outside the earth)
Can you exactly give a picture (drawing for this ). I think I cannot imagine this.

 

[Hyperspace2]

Mathematically it is related. The differences are:
1) the surface described would be a curved space embedded in a higher-dimensional flat space whereas there is no embedding space considered or required in GR
2) the surface described would be Riemannian, but curved spacetime is pseudo-Riemannian
3) the surface described is 2 dimensional whereas it is 1+3 dimensional for GR

I don't have any idea of embedded space, riemannian,curved reminan, pseudo-reminan
could you explain (or give me a link)

 

[pervect]

One can think of general relativity as drawing the space-time diagrams of special relativity on a curved surface. Though people don't actually do this except as a teaching exercise. There's a paper by Marolf I could quote if someone is interested, though it's probably too advanced for the original poster. One particular surface is derived (it's nothing simple like a sphere) which has the correct properties to be a slice of space-time around a black hole or other massive body (specifically, the r-t plane of the Schwarzschild geometry).

 

[HallsofIvy]

Oh sorry do we see a circular or rectangular ground (I am little confused now)
Thanks for reply
Let me conclude
The rectangular ground(or circular ground) we consider would be actually a curved incomplete equilateral triangle .

You were the one who said "rectangular". This is the first time you have mentioned a "horizon"! Exactly what the horizon would look like depends upon how big your sphere is.

You people mean that the locus of the horizon that observer see actually end up being curved equilateral triangle (as seen from outside the earth)
Can you exactly give a picture (drawing for this ). I think I cannot imagine this.

No, no one has said anything about the "locus of the horizon" because we did not know you were talking about a "horizon". You started by talking about a "never ending football ground" which implied that you intended a rectangle.

All of this has, as said before, nothing to do with General Relativity which is about the dynamics of motion in a gravitational field, not about what "horizons" look like.

 

[ghwellsjr]

...
The rectangular ground(or circular ground) we consider would be actually a curved incomplete equilateral triangle .

You people mean that the locus of the horizon that observer see actually end up being curved equilateral triangle (as seen from outside the earth)
Can you exactly give a picture (drawing for this ). I think I cannot imagine this.

If you look at the wikipedia.org opening page, you will see a sphere made out of "rectangular" pieces (with connecting tabs). You will note that you cannot see the pieces on the bottom of the sphere (they are out of view) and the pieces that should go on the top of the sphere are missing. Now why do you suppose they are missing? Well, it's because if they attempted to draw them in, it would be obvious that you cannot make a sphere out of just rectangular pieces. At the top, you would have to have some other shape, like triangular pieces or a round piece to make it fit.

If you image these pieces seem flat to a tiny bug walking on the surface, he could observe that his map of the world does not consist of only a bunch of "rectangular" pieces. Somewhere, they would have to be a different shape and from that he would conclude that his "world" is not flat but is in a larger dimension that is curved in some way, even though he cannot detect the curvature of his world.

I think you understand that the equilateral triangle that I mentioned earlier is not really a flat equilateral triangle, it only seems flat to a being on the surface of this immense globe, and it is curved just like the puzzle pieces on the wikipedia page, but if it is the correct size with each side being 1/4 the circumference of the globe, then it would not be a curved incomplete equilateral triangle, but rather a curved complete equilateral triangle.

But these are just examples of how we on the surface of our perfectly smooth globe called the Earth could determine that it was not flat but curved by measuring the angles of any large enough triangle and seeing that the sum of the angles exceeds 180 degrees. We prefer explaining this with triangles because you can always make a complete triangle of any size on any globe (although it will not generally be an equilateral triangle), whereas if you attempt to make a rectangle with 90 degree corners, it will always be incomplete (except in some weird cases).