Hi !
I find all the textbooks showing spacetime as a 2D plane and the heavy
objects bending it.
but how does this be interpreted in the real 3D world?

fuzzlogic wrote:
I find all the textbooks showing spacetime as a 2D plane and the heavy
objects bending it.

That is a mere ANALOGY. For instance, it cannot possibly "explain"
gravitation, because it requires gravity to make the heavy objects bend
the "rubber sheet".


but how does this be interpreted in the real 3D world?

It needs to be applied in 4D spaceTIME. The word "curvature" is really a
metaphor, or rather, a technical word with a specific meaning not
applicable to our everyday lives (unless you are a physicist (:-)).

The justification for using this term is that mathematically what is
called "curvature" in N-dimensional manifolds has a direct relationship
to the local radius of curvature for a 2d surface.

To understand this requires some modest amount of study. I suggest:

Geroch, _General_Relativity_from_A_to_B_. This is a
non-mathematical introduction to the concepts of GR.


Tom Roberts

fuzzlogic wrote:
Hi !
I find all the textbooks showing spacetime as a 2D plane and the heavy
objects bending it.
but how does this be interpreted in the real 3D world?

All such analogies are meant to illustrate the difference between
*interior* properties of a space and *exterior* properties of a space.
As an example of that, a 2D sphere (like a globe) embedded in 3D space
is supposed to teach you that parallel lines on a 2D surface *can*
intersect. The common reaction to that is to say, "But that's because
the surface is curved in a higher dimensional (3D) space)." The
counter-response is to say, "Yes, but there are things that you can
deduce about that surface *without* resorting to looking at it from the
perspective of the embedding space. When we do that, we are looking at
the *interior* properties, not the *exterior* properties of that
space."
The lesson then is to focus on the interior properties only. When you
do that, then it is easier to understand how those properties apply to
a curved 3D space or even a curved 4D space, without having to try to
visualize an embedding 4D or 5D space (which is hard).

PD

fuzzlogic wrote:
Hi !
I find all the textbooks showing spacetime as a 2D plane and the heavy
objects bending it.
but how does this be interpreted in the real 3D world?

if you cant map the visualization of the 2d to a an evt existent 3d
then
maybe you should go outside and find another hoby

PD wrote:
fuzzlogic wrote:
Hi !
I find all the textbooks showing spacetime as a 2D plane and the heavy
objects bending it.
but how does this be interpreted in the real 3D world?

All such analogies are meant to illustrate the difference between
*interior* properties of a space and *exterior* properties of a space.
As an example of that, a 2D sphere (like a globe) embedded in 3D space
is supposed to teach you that parallel lines on a 2D surface *can*
intersect. The common reaction to that is to say, "But that's because
the surface is curved in a higher dimensional (3D) space)." The
counter-response is to say, "Yes, but there are things that you can
deduce about that surface *without* resorting to looking at it from the
perspective of the embedding space. When we do that, we are looking at
the *interior* properties, not the *exterior* properties of that
space."
The lesson then is to focus on the interior properties only. When you
do that, then it is easier to understand how those properties apply to
a curved 3D space or even a curved 4D space,

whay confussing people more, what 4d, he asked about 3d,
what 4d, are you eating sandwitches again?

without having to try to
visualize an embedding 4D or 5D space (which is hard).

PD

"Tom Roberts" kirjoitti viestissä
t...
fuzzlogic wrote:
I find all the textbooks showing spacetime as a 2D plane and the heavy
objects bending it.

That is a mere ANALOGY. For instance, it cannot possibly "explain"
gravitation, because it requires gravity to make the heavy objects bend
the "rubber sheet".


but how does this be interpreted in the real 3D world?

It needs to be applied in 4D spaceTIME. The word "curvature" is really a
metaphor, or rather, a technical word with a specific meaning not
applicable to our everyday lives (unless you are a physicist (:-)).

The justification for using this term is that mathematically what is
called "curvature" in N-dimensional manifolds has a direct relationship
to the local radius of curvature for a 2d surface.

To understand this requires some modest amount of study. I suggest:

Geroch, _General_Relativity_from_A_to_B_. This is a
non-mathematical introduction to the concepts of GR.

Tom Roberts

Curvature of space is not a metaphor. It is a correct description of
reality. But space time is a metaphor. It is a wrong explanation.

http://www.wakkanet.fi/~fields/

Henry Haapalainen

"Henry Haapalainen" wrote in message
...

"Tom Roberts" kirjoitti viestissä
t...
fuzzlogic wrote:
I find all the textbooks showing spacetime as a 2D plane and the heavy
objects bending it.

That is a mere ANALOGY. For instance, it cannot possibly "explain"
gravitation, because it requires gravity to make the heavy objects bend
the "rubber sheet".


but how does this be interpreted in the real 3D world?

It needs to be applied in 4D spaceTIME. The word "curvature" is really a
metaphor, or rather, a technical word with a specific meaning not
applicable to our everyday lives (unless you are a physicist (:-)).

The justification for using this term is that mathematically what is
called "curvature" in N-dimensional manifolds has a direct relationship
to the local radius of curvature for a 2d surface.

To understand this requires some modest amount of study. I suggest:

Geroch, _General_Relativity_from_A_to_B_. This is a
non-mathematical introduction to the concepts of GR.

Tom Roberts

Curvature of space is not a metaphor. It is a correct description of
reality. But space time is a metaphor. It is a wrong explanation.


Since it is fully in accord with all experimental evidence your claim lacks
any foundation.

Bill



http://www.wakkanet.fi/~fields/

Henry Haapalainen

fuzzlogic wrote:
Hi !
I find all the textbooks showing spacetime as a 2D plane and the heavy
objects bending it.
but how does this be interpreted in the real 3D world?

FYI ....

A 2-d space needs a 3-d "embedding space" (1 extra dimension) to be
able to curve. It's easy to show this in textbooks. But ....

A 3-d space needs a 6-d embedding space to do the same. And a 4-d
space needs a 10-d embedding space to curve. Not much good for drawing
in textbooks.

And in general, you'll be galvanized to know, an n-dimensional space
needs an n(n+1)/2 dimensional embedding space to curve.

Not that the mathematical physicists imagine the embedding spaces
themselves are real (if they are, what's stopping us getting to them?).
They just deal with the *internal* geometrical properties of a given
space, which is equivalent to external curving.


.....Oh well, perhaps they didn't need those embedding spaces after all!

Mahmoud In My Dinner Jacket wrote:
[...]

All your values for the dimensionality of an embedding space are wrong.
For instance, not all 2-d manifolds can be embedded in a 3-d space --
the Klein bottle is an explicit counterexample (and it is flat).

In fact, to isometrically embed an arbitrary 3,1-dimension spacetime
manifold in a FLAT manifold requires a manifold of k,m dimensions (k
spacelike and m timelike). The tightest limits on k and m I know of a
k=88, m=2 -- that is ENORMOUS compared to your guesses.


....Oh well, perhaps they didn't need those embedding spaces after all!

GR does not depend on tham in any way.


Tom Roberts

"Bill Hobba" kirjoitti viestissä
...

"Henry Haapalainen" wrote in message
...

"Tom Roberts" kirjoitti viestissä
t...
fuzzlogic wrote:
I find all the textbooks showing spacetime as a 2D plane and the
heavy
objects bending it.

That is a mere ANALOGY. For instance, it cannot possibly "explain"
gravitation, because it requires gravity to make the heavy objects bend
the "rubber sheet".


but how does this be interpreted in the real 3D world?

It needs to be applied in 4D spaceTIME. The word "curvature" is really
a
metaphor, or rather, a technical word with a specific meaning not
applicable to our everyday lives (unless you are a physicist (:-)).

The justification for using this term is that mathematically what is
called "curvature" in N-dimensional manifolds has a direct relationship
to the local radius of curvature for a 2d surface.

To understand this requires some modest amount of study. I suggest:

Geroch, _General_Relativity_from_A_to_B_. This is a
non-mathematical introduction to the concepts of GR.

Tom Roberts

Curvature of space is not a metaphor. It is a correct description of
reality. But space time is a metaphor. It is a wrong explanation.

http://www.wakkanet.fi/~fields/

Henry Haapalainen

Since it is fully in accord with all experimental evidence your claim
lacks
any foundation.

Bill

Don't you ever get tired on that? HH