Curved Space: Is it More Than a Metaphor?

[xMonty]

Is space really curved/warped in presence of mass or is it just a metaphore, is it more than a mathematical concept? i also read space is denser near the mass is this oversimplification.

Also light bends around mass but that can also be explained as light has energy so it falls like everything else.

 

[tiny-tim]

Hi xMonty!

Is space really curved/warped in presence of mass or is it just a metaphore, is it more than a mathematical concept?

It's more of a metaphor … there's no actual higher-dimensional space in which our space is emebedded … if there were, our space would really be curved …

but since our space is "all there is", curvature can only be defined mathematically.

… i also read space is denser near the mass is this oversimplification.

I'm not sure what "denser" would mean … which book is that from, and what's the actual quotation?

Also light bends around mass but that can also be explained as light has energy so it falls like everything else.

Light has energy and momentum, so it follows trajectories which depend on energy and momentum, like everything else.

You can find those trajectories by putting the same line-element equation, c²dt² - dx² = m², into the space-time metric, but of course with m = 0.

 

[A.T.]

Is space really curved/warped in presence of mass or is it just a metaphore, is it more than a mathematical concept?

It is a mathematical model.

i also read space is denser near the mass is this oversimplification.

Varying density is one way to visualize intrinsic curvature. The other is embedding in higher dimensional non-curved manifolds. This post explains the two methods:
https://www.physicsforums.com/showthread.php?p=2003340

Also light bends around mass but that can also be explained as light has energy so it falls like everything else.

Light does fall like everything else. For a stationary observer in a g-field everything including light seems to be accelerated locally at the same rate. That is an effect of time-curvature. But you don't get the correct trajectory of an moving object from that local acceleration alone. The overall observed trajectory is also affected by the spatial curvature. Here some pictures:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

 

[xMonty]

Hi xMonty!


It's more of a metaphor … there's no actual higher-dimensional space in which our space is emebedded … if there were, our space would really be curved …

but since our space is "all there is", curvature can only be defined mathematically.


I'm not sure what "denser" would mean … which book is that from, and what's the actual quotation?


Light has energy and momentum, so it follows trajectories which depend on energy and momentum, like everything else.

You can find those trajectories by putting the same line-element equation, c²dt² - dx² = m², into the space-time metric, but of course with m = 0.

If space can't be curved then what does it mean when they say that cosmologists have found k=1 and they think that the universe is so large that's why in our observable universe k=1 (i.e. space is flat just like the Earth's surface is flat for small distances)

 

[tiny-tim]

If space can't be curved then what does it mean when they say that cosmologists have found k=1 and they think that the universe is so large that's why in our observable universe k=1 (i.e. space is flat just like the Earth's surface is flat for small distances)

Space can be curved in the mathematically-defined metaphorical sense.

Observation has shown that, on a large-scale, it isn't curved ("Analysis of data from WMAP confirmed the universe is flat with only a 2% margin of error" … see http://en.wikipedia.org/wiki/Shape_of_the_universe" ), but of course it still is curved locally, ie near any particular mass.

 

[xMonty]

Space can be curved in the mathematically-defined metaphorical sense.

Observation has shown that, on a large-scale, it isn't curved ("Analysis of data from WMAP confirmed the universe is flat with only a 2% margin of error" … see http://en.wikipedia.org/wiki/Shape_of_the_universe" ), but of course it still is curved locally, ie near any particular mass.

So Space is curved near huge bodies?

 

[tiny-tim]

bubble-warp!

So Space is curved near huge bodies?

Yes, of course …

like corrugated iron, or bubble-wrap, which is curved locally but flat on the large-scale.

Space is bubble-warp … pop it, and you get black holes! ​

 

[xMonty]

Yes, of course …

like corrugated iron, or bubble-wrap, which is curved locally but flat on the large-scale.

Space is bubble-warp … pop it, and you get black holes! ​

But above people are saying that "its just a mathematical model"

 

[HallsofIvy]

Yes, just as F= ma or F= GmM/r² are mathematical models. But you will still fall down if you trip!

 

[Cleonis]

Is space really curved/warped in presence of mass or is it just a metaphore, is it more than a mathematical concept?

As previous posters mentioned, it's important to distinguish between intrinsic curvature and extrinsic curvature.

The coordinate grid of longitude lines and latitude lines that is defined on the Earth is an example of a coordinate grid with extrinsic curvature.
Note that if you were to take the Earth's surface, and treat it as if it's a flat surface, then you will run into discrepancies: you will find that if you draw a perfect circle and you measure both circumference and diameter along the Earth's surface, then if the circle is large enough you will find that the ratio of diameter and circumference is not pi, but some other number.

We live in a world with 3 spatial dimensions, so any flat surface, embedded in 3D space can be curved.

Somewhat counterintuitively, this embedding in a space with more dimensions is not in itself necessary to enable curvature. (to find examples you'll need to do some googling with the expressions 'intrinsic curvature' and 'extrinsic curvature'.

Getting to your question: if a region of spacetime is sufficiently curved then spatially you will find the same kind of deviation from pi.

I recall reading (I don't remember where) that the deformation of space around the Earth is such that the ratio of Earth diameter and Earth circumference will not be exactly pi. If memory serves me the deviation is in the order of milimeters, but don't quote me on that. Likewise the ratio of volume to surface area will not be the Euclidean one.

I suppose the above considerations played an important role in why the metaphor 'spacetime curvature' has become the most widely used. Still, it's better to be cautious, and keep thinking of it as a metaphor.

Cleonis