Adding Extra Dimension for Calculations in Curved Spaces

In summary, the author concludes that adding a virtual dimension to the space does not help with calculations.
  • #1
Kairos
182
16
as calculations are technically difficult in curved spaces, I wonder if we would obtain the same results by adding one additional (virtual) dimension in order to embed the space in a higher order Euclidean volume, just to facilitate the treatments? (for example embed a 3D hypersphere in a 4D euclidean space)
thank you in advance
 
Physics news on Phys.org
  • #2
Does this become easier, because it is on a flat plane?

245390


You can calculate with spacetime as if it was flat, at least either locally or on a global scale. Since the embedding doesn't change the geometric shape, there will be no gain in doing so.
 
  • Like
Likes vanhees71 and Kairos
  • #3
The entire point of differential geometry is not to have to rely on an embedding space because it generally complicates things.
 
  • Like
Likes vanhees71 and Kairos
  • #4
Kairos said:
as calculations are technically difficult in curved spaces, I wonder if we would obtain the same results by adding one additional (virtual) dimension in order to embed the space in a higher order Euclidean volume, just to facilitate the treatments? (for example embed a 3D hypersphere in a 4D euclidean space)
thank you in advance
Embedding curved spaces in flat higher dimensionless spaces is usually done to help with visualization. But for calculations, you usually want as few variables and as little redundancy as possible.
 
  • Like
Likes Kairos
  • #5
OK thanks
I supposed that the simplicity of euclidean rules would compensate the additional spatial variable. bad idea !
 
  • #6
Kip Thorne has written some popularizations in "Interstellar" about the embedding approach. But I haven't seen any non-popularized treatment of General Relativity using embeddings.

So, if you're attracted to the approach and you don't mind reading popularizations (which are usually limited, even when well written), you could try "Interstellar", but for a serious, textbook study you'd want to learn the differential geometry approach.

There's at least one other approach to GR, that uses funky fields that warp rulers and clocks in a flat space-time. This is akin to Einstein's discussion of rulers on a heated marble slab as a model for non_euclidean spatial geometries. This approach has been outlined by Straumann in "Reflections on Gravity" <<link>>. I rather suspect that Straumann's approach has some limits in regards to modelling some of the topological features that the full theory handles, and that this matters in such topics as understanding black holes, but the author doesn't discusss these limitations, unfortunately.
 
  • Like
Likes Kairos

FAQ: Adding Extra Dimension for Calculations in Curved Spaces

1. What is the purpose of adding extra dimensions for calculations in curved spaces?

The purpose of adding extra dimensions is to account for the curvature of space in certain physical systems, such as in general relativity. By adding extra dimensions, we can better describe and understand the behavior of these systems.

2. How many extra dimensions are typically added for these calculations?

The number of extra dimensions added can vary depending on the specific system being studied. In general relativity, for example, four dimensions are typically added to account for the three dimensions of space and one dimension of time.

3. How do these extra dimensions affect the calculations?

These extra dimensions affect the calculations by allowing for a more accurate representation of the curvature of space. By including these dimensions, we can better understand the behavior of particles and objects in curved spaces.

4. Are these extra dimensions observable?

No, these extra dimensions are not observable in our everyday experiences. They are considered to be "compactified" or "curled up" dimensions that are too small for us to detect directly. However, their effects can be observed through their influence on the behavior of particles and energy in curved spaces.

5. How does the addition of extra dimensions impact our understanding of the universe?

The addition of extra dimensions has greatly expanded our understanding of the universe, particularly in the field of theoretical physics. It has allowed us to develop more accurate and comprehensive models of the universe, such as the theory of general relativity, and has opened up new areas of research and exploration into the nature of space and time.

Similar threads

Back
Top