Approximating smooth curved manifolds with "local bits" of curvature?

[Spinnor]

Consider the electric and magnetic fields around a dipole antenna,

Suppose these fields represent some type of curvature in space and time. Suppose where the fields are strong we have greater curvature. Also suppose these fields are really some very large but finite sum of "moving local bits" of curvature whose sum closely represents the classical fields above. So the question is can we get smooth curved manifolds from small local bits of curvature if we don't look too closely, (any small volume we might look at would contain many bits whose curvature overlaps and averages). Is there a branch of mathematics that might deal with this?

Thanks.

 

[docnet]

Edit:

If I'm understanding right, you can numerically model such a system as a function of space and time by assigning time-dependent vectors to predetermined 3-d grid points. A loose motivation is to evaluate the points as vectors of a vector field on a higher dimensional manifold. The grid spacing can be as fine as you want within reason, but you can't cover with a finite number of points.

Consider the electric and magnetic fields around a dipole antenna,

View attachment 344716

View attachment 344717
Suppose these fields represent some type of curvature in space and time. Suppose where the fields are strong we have greater curvature. Also suppose these fields are really some very large but finite sum of "moving local bits" of curvature whose sum closely represents the classical fields above. So the question is can we get smooth curved manifolds from small local bits of curvature if we don't look too closely, (any small volume we might look at would contain many bits whose curvature overlaps and averages). Is there a branch of mathematics that might deal with this?

Thanks.

If I'm understanding right, you can numerically model such a system as a function of space and time by assigning time-dependent vectors to predetermined 3-d grid points. A loose motivation is to evaluate the points as components of a vector field of higher dimensional manifold. The grid spacing can be as fine as you want within reason, but you can't cover ##\mathbb{R}^3## with a finite number of points.

Explain 'moving local bits'.

 

[docnet]

My reply disappeared because the site is glitchy on my phone. I'll post a mother reply later

 

[Spinnor]

If I'm understanding right, you can numerically model such a system as a function of space and time by assigning time-dependent vectors to predetermined 3-d grid points. A loose motivation is to evaluate the points as components of a vector field of higher dimensional manifold. The grid spacing can be as fine as you want within reason, but you can't cover ##\mathbb{R}^3## with a finite number of points.

Explain 'moving local bits'.

I am having trouble coming up with a clear answer so will have to think a bit more. Thanks.

 

[Spinnor]

Again consider the electric fields surrounding an antenna at an instant of time.

We know that an antenna emits energy, suppose the energy is in the form of a very large but finite number of particles which move away from the antenna at the speed of light. Google "how many photons does a typical radio signal emit" the result begins,

• 1 Watt transmitter at 100 MHz: Emits about 1E25 photons per second, or about 100 billion photons per square meter of reception area per second at a range of 60 miles
• 1 kW radio transmitter at 800 Hz: Emits 1.88 × 10³³ photons per second
• 84-kW AM radio station at 1000 kHz: Emits 1.272 x 10^32 photons per second
• 940 kHz radio station: Emits 4 × 10^31 photons per second
• FM radio station at 98.1 MHz: Emits 7.7 × 10^29 photons per second
  

Suppose that associated with each particle is a wavelength inversely proportional to their energy. We might also associate a helicity with each particle, of either left or right handed type.

Now the problem is, can we associate electric and magnetic fields with each particle such that the square of these fields (which is a measure of energy density) is significant only in a volume of order the wavelength cubed centered on the particle. When there are very many particles emitted per second from the antenna we want the fields of each particle to sum to the classical fields shown above.

The electric and magnetic fields far from the antenna have the property that the divergence of the fields is zero, the field lines form closed loops. Let us assume the fields associated with each particle also have this property that their divergence is zero, local fields of zero divergence sum to global fields of zero divergence (local used in the sense that the energy and thus the fields of a particle is mostly confined to a volume of order the wavelength cubed).

The moving local bits are then the wavelength sized regions of electric and magnetic fields that move at the speed of light.

One might want to consider how fields look in a frame of reference that moves away or towards the antenna with constant velocity. Relativity theory constrains our ideas. For example, relativity tells us that there is a velocity we can move away from the antenna such that particles that move right past us will have their energy halved and their wavelength doubled, red-shift. One might ask if the fields we assigned to each particle in our rest frame transform in the right way according to the special theory of relativity in a moving frame of reference.

To move forward I think one needs to just guess at some approximate set of fields and examine if this approximation makes any sense at all.

Thanks for your suggestions.