Electromagnetic Fields and the manipulation of Space-Time

[Dinoduck94]

TL;DR Summary

Mathematical and Physical queries in regards to Electromagnetic Fields and their manipulation of Space-Time.

Summary: Mathematical and Physical queries in regards to Electromagnetic Fields and their manipulation of Space-Time.

I recently started looking into Einstein's Field Equations, to get a better understanding of how mass distorts and curves the plane of Space-Time, however from doing this I understood that it isn't actually mass that distorts Space-Time but Energy - which is directly proportional to the mass of an object (E=mc²).

So from this, Energy creates the sensation of Gravity, rather than the mass of an Object.
This made me think that if a large enough magnetic field was generated, that the Energy held within the magnetic field could produce a Gravitational effect - so I tried doing the calculations to see what figures we would be looking at in regards to this...

We know that the Earth's Gravitation Acceleration is about 9.81 m/s²
This is calculated from:

a = mG / d²
Where a is Gravitational Acceleration in m/s²; m is the mass in Kg; G is the Gravitational Constant of 6.67408x10^-11; and d is the distance from the object.
We can re-arrange this to solve for m:
m = ad² / G

Because we know mass is directly proportional with Energy from E=mc², we can substitute this into the equation above:
E = (ad² / G)c² ∴ E = ac²d² / G

Solving for 'a' we get:
a = EG / c²d²

If we were to make a coil to generate the Magnetic field we needed, then we can use the below equation to find the Induced Energy (E) from the Inductance (L) and the Current (I) on the coil:

E = 1/2 LI²

If we substituted the above formula for E in our previous formula, then we get this:

a = GLI² / 2c²d²

If we were to make the coil, we would need to calculate the amount of Inductance generated; using this equation we can do that:

L = N²∙μ₀∙μ_r∙(D/2) ∙ (ln(8D/d)-2)

Where N is the number of turns in the coil; μ₀ is the Permeability of free space; μ_r is the Relative Permeability of the Core; D is the diameter of each coil; and 'd' is the diameter of the cable used. I got this formula https://www.eeweb.com/tools/coil-inductance?utm_source=AspenCore&utm_medium=ELECTRONICS%20TUTORIALS

If we were to use a 99.85% pure Iron core, with a Relative Permeability of 200,000; each coil being 0.2m in diameter with 50mm² cable for 400A of current; and assuming that we would be standing 1m away from it - then we can start putting this equation together.

Unfortunately this is where it gets ridiculous - using all of the above, to get a meaningful Gravitation Acceleration; N has to be an obscenely large number... in this case 600,000,000,000 coils...

L = 600,000,000,000²∙1.26x10^-6∙200,000∙(0.2/2) ∙ (ln(8∙0.2/(2∙√(50/π))-2) = 2.98664x10²² H
a = 6.67408x10^-11∙2.98664x10²²∙400² / 2∙299792458²∙1²
a = 1.77m/s²

This is slightly more than the gravitational pull of the moon - 1.62m/s²

Now...

This is the mathematical side of it, practically speaking that would never work for multiple reasons but perhaps mainly because the vast amount of cable you'd actually need to do this (enough for 50mm² cable to go around the Earth 9.5 million times).

This is why I have come on here to discuss it, so; first of all - is my maths correct?
Second of all, is there any current technology or method that can be used to increase Inductance without having to do 600 billion coils?
Is there a material with a higher Relative Permeability than the 99.85% Pure Iron? I found this 200,000 value after googling it, so not sure if it correct!

My last question, is more ignorant than the others - I have heard that it is possible to get Negative Inductance - I know nothing about it, but the calculations show Gravitational Acceleration going negative when there is Negative Inductance - would this "straighten out" the existing curvature of Space-Time, nulling Gravity?

Thanks!

 

[.Scott]

The only way energy is known to produce gravity is through its mass equivalence. In the case of you electric coils, the coils themselves would provide more mass (and thus gravity) than the energy it is holding.
Negative inductance is just an electronic concept - like resistance or capacitance. It would have no effect on gravity.

 

[Dinoduck94]

The only way energy is known to produce gravity is through its mass equivalence. In the case of you electric coils, the coils themselves would provide more mass (and thus gravity) than the energy it is holding.
Negative inductance is just an electronic concept - like resistance or capacitance. It would have no effect on gravity.

Hi Scott, thanks for answering about the Negative Inductance, however I believe you are wrong about energy only producing gravity through its mass equivalence. Einstein's General Relativity says that any form of energy is a source of gravity. While you would still need an obscene amount of Energy to match the gravitational effect of actual mass, the relationship is still there.

 

[ZapperZ]

Summary: Mathematical and Physical queries in regards to Electromagnetic Fields and their manipulation of Space-Time.

I recently started looking into Einstein's Field Equations, to get a better understanding of how mass distorts and curves the plane of Space-Time, however from doing this I understood that it isn't actually mass that distorts Space-Time but Energy - which is directly proportional to the mass of an object (E=mc²).

So from this, Energy creates the sensation of Gravity, rather than the mass of an Object.
This made me think that if a large enough magnetic field was generated, that the Energy held within the magnetic field could produce a Gravitational effect - so I tried doing the calculations to see what figures we would be looking at in regards to this...

We know that the Earth's Gravitation Acceleration is about 9.81 m/s²
This is calculated from:

a = mG / d²
Where a is Gravitational Acceleration in m/s²; m is the mass in Kg; G is the Gravitational Constant of 6.67408x10^-11; and d is the distance from the object.
We can re-arrange this to solve for m:
m = ad² / G

Because we know mass is directly proportional with Energy from E=mc², we can substitute this into the equation above:
E = (ad² / G)c² ∴ E = ac²d² / G

Solving for 'a' we get:
a = EG / c²d²

If we were to make a coil to generate the Magnetic field we needed, then we can use the below equation to find the Induced Energy (E) from the Inductance (L) and the Current (I) on the coil:

E = 1/2 LI²

If we substituted the above formula for E in our previous formula, then we get this:

a = GLI² / 2c²d²

If we were to make the coil, we would need to calculate the amount of Inductance generated; using this equation we can do that:

L = N²∙μ₀∙μ_r∙(D/2) ∙ (ln(8D/d)-2)

Where N is the number of turns in the coil; μ₀ is the Permeability of free space; μ_r is the Relative Permeability of the Core; D is the diameter of each coil; and 'd' is the diameter of the cable used. I got this formula https://www.eeweb.com/tools/coil-inductance?utm_source=AspenCore&utm_medium=ELECTRONICS%20TUTORIALS

If we were to use a 99.85% pure Iron core, with a Relative Permeability of 200,000; each coil being 0.2m in diameter with 50mm² cable for 400A of current; and assuming that we would be standing 1m away from it - then we can start putting this equation together.

Unfortunately this is where it gets ridiculous - using all of the above, to get a meaningful Gravitation Acceleration; N has to be an obscenely large number... in this case 600,000,000,000 coils...

L = 600,000,000,000²∙1.26x10^-6∙200,000∙(0.2/2) ∙ (ln(8∙0.2/(2∙√(50/π))-2) = 2.98664x10²² H
a = 6.67408x10^-11∙2.98664x10²²∙400² / 2∙299792458²∙1²
a = 1.77m/s²

This is slightly more than the gravitational pull of the moon - 1.62m/s²

Now...

This is the mathematical side of it, practically speaking that would never work for multiple reasons but perhaps mainly because the vast amount of cable you'd actually need to do this (enough for 50mm² cable to go around the Earth 9.5 million times).

This is why I have come on here to discuss it, so; first of all - is my maths correct?
Second of all, is there any current technology or method that can be used to increase Inductance without having to do 600 billion coils?
Is there a material with a higher Relative Permeability than the 99.85% Pure Iron? I found this 200,000 value after googling it, so not sure if it correct!

My last question, is more ignorant than the others - I have heard that it is possible to get Negative Inductance - I know nothing about it, but the calculations show Gravitational Acceleration going negative when there is Negative Inductance - would this "straighten out" the existing curvature of Space-Time, nulling Gravity?

Thanks!

If this doesn't break any rules, this should go into the Relativity forum.

Zz.

 

[Dinoduck94]

If this doesn't break any rules, this should go into the Relativity forum.

Zz.

Hi ZapperZ,

Should I post it again in the Relativity forum, or is there a way of moving it?

Thanks

 

[ZapperZ]

Hi ZapperZ,

Should I post it again in the Relativity forum, or is there a way of moving it?

Thanks

Do not post it again or you'll definitely break the forum rules for double/multiple posting. Wait for the Moderators to move it. They have been alerted.

Zz.

 

[PeterDonis]

it isn't actually mass that distorts Space-Time but Energy

No, it's the stress-energy tensor. That includes energy (more precisely, energy density), but it also includes other things.

This made me think that if a large enough magnetic field was generated, that the Energy held within the magnetic field could produce a Gravitational effect

The electromagnetic field's stress-energy tensor is well understood:

https://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor
As a rough order of magnitude, the terms will be proportional to the squares of the fields. In practical terms, any field that can be produced by our current technology will have far too small a stress-energy tensor for its effect on spacetime curvature ("gravitational effect") to be detectable.

I tried doing the calculations

It would be better to look at the actual stress-energy tensor of an electromagnetic field and see how it is calculated, and use that.

I have heard that it is possible to get Negative Inductance

"I have heard" is not helpful. Please give a specific reference.

the calculations show Gravitational Acceleration going negative when there is Negative Inductance

That's because your calculations are just substituting "energy" into some formulas and waving your hands. As above, you should instead look at the actual stress-energy tensor of the electromagnetic field.

would this "straighten out" the existing curvature of Space-Time, nulling Gravity?

No.

 

[pervect]

It would be better to look at the actual stress-energy tensor of an electromagnetic field and see how it is calculated, and use that.

I could use a sanity check here:

Using <wiki link>, I get for the stress energy tensor of just a field only in the $B_x$ direction
$$
T^{\mu\nu} = \begin{bmatrix} \rho & 0 & 0 & 0 \\ 0 & -\rho & 0 & 0 \\ 0 & 0 & \rho & 0 \\ 0 & 0 & 0 & \rho \end{bmatrix}
$$

where $\rho = \frac{B^2}{2\mu_0}$

So if we use the Komar mass formula where we sum the diagonals we get $\rho -\rho + \rho + \rho = 2 \rho$.

This would suggest a 2:1 error, but I rather suspect that there are physical pressure terms in addition to the stress-energy terms of the magnetic field itself that must exist within whatever's generating the magnetic field. For instance, the fermi lab magnets that exploded because of the physical stress in the wires illustrates that there are physical pressures in electromagnets.

Thus, I'd think the pressure terms would probably cancel out, otherwise we'd have a paradox where converting chemical energy into electricity and then a magnetic field would change the mass of the system at infinity which is impossible.

In any event, I don't think there'd be much more than a 2:1 error in the naieve approach that the OP used computed the stored energy in the magnetic field and assuming it acted like mass. And probably no error at all.

I don't know why the OP thinks that it's possible to have negative energy stored in a magnetic field, though - that doesn't seem to be supported by the equations or very physical.

 

[PeterDonis]

I get for the stress energy tensor of just a field only in the $B_x$ direction...

Yes, which shows that, as I said in post #7, the SET is proportional to the squares of the fields, so in this case it's proportional to $B_x^2$. And if you run the numbers for any value of $B_x$ that's within the ability of our technology to produce now or in the foreseeable future, you will see that the SET due to the magnetic field is miniscule compared to, for example, the rest energy density of the material in the magnet. And even more miniscule compared to the amount of stress-energy it takes to produce detectable spacetime curvature.

the naieve approach that the OP used computed the stored energy in the magnetic field

The OP's approach was not to take the squares of the fields and use that as the estimate. If it had been, he would not have been led to an erroneous idea about "negative energy". See below.

I don't know why the OP thinks that it's possible to have negative energy stored in a magnetic field

Because he wasn't taking the simple approach you took of actually looking at the stress-energy tensor of an EM field, seeing that the components proportional to the squares of the fields, looking at how those components combine in something like a Komar mass, and therefore understanding that the overall "energy" is going to be positive. Instead he was waving his hands with formulas using inductance.

 

[pervect]

Because he wasn't taking the simple approach you took of actually looking at the stress-energy tensor of an EM field, seeing that the components proportional to the squares of the fields, looking at how those components combine in something like a Komar mass, and therefore understanding that the overall "energy" is going to be positive. Instead he was waving his hands with formulas using inductance.

I did notice that while the OP stated

I recently started looking into Einstein's Field Equations,

he never actually wrote down any of the actually field equations, which would be of course $G^{\mu\nu} =$(some constant) $T^{\mu\nu}$.

The value of some constant depends on the unit choice, I'm very used to geometric units so I'd have to look it up to get the value of said constant for SI units.

I thought it was interesting and possibly educational to write down the stress energy tensor $T^{\mu\nu}$ which, as you suggested, is the first step in figuring out what Eintein's field equations actually say about the problem.

The arguments presented by the OP didn't actually use Einstein's field equations at all as near as I could tell from my read through.

In spite of not actually using the field equations, I think the OP's hand waving turned out to give the correct answer. At least in broad outlines, I did not plug in the actual numbers or check the arithmetic.

It could possibly be off by a factor of two (though I have reasons to suspect otherwise. It'd probably get too confusing to go into details). A factor of 2:1 isn't going to make any difference to the conclusion, which hopefully is shared by everyone at this point, that the effect of the energy stored in an inductor on it's mass is undetectable with present technology.

Writing $\frac{1}{2} L I^2$ for the stored energy gives the same result as integrating the energy density in the electromagnetic field (which is proportiaonal to B^2 as you state), as it must.

The bit about "negative inductance", I'd agree, came from left field.