Our analysis of the electric field is based on the energy which is freed or absorbed when we move q closer to Q or away from Q. This energy is completely analogous for weak fields between m and M. We should see analogous phenomena.
The electromagnetic / gravity field of a rotating disk
Ever since our post on August 29, 2023, we have been confused about the gravity field of a mass flow. A typical example of a mass flow is a rotating disk.
<-- ω
_____
/ \
| | +
\______/
+
^ V
|
• q = e-
Let us consider a rotating electrically positively charged disk. A negative test charge q approaches it.
The disk is uniformly charged. Let us superpose a static disk onto it, such that the static disk is uniformly charged and cancels the positive charge of the rotating disk in the laboratory frame.
We now essentially have in the disk configuration many wire elements dl creating a magnetic field B. We can calculate the magnetic field of the rotating disk using the Biot-Savart law.
The electric field of the positively charged rotating disk is the opposite of the electric field of the static negatively charged disk.
The 16X power of a radiating gravity quadrupole: gravity is NOT analogous to electromagnetism
If the magnetic effect of a (weak) gravity field is totally analogous to electromagnetism, then we above have a recipe for calculating the gravitomagnetic field of a moving mass.
But the Kerr metric and various other literature claims that the gravitomagnetic effect of is 4X compared to the analogous electromagnetic effect.
Also, we know from binary pulsars that a gravity quadrupole radiates 16X the energy of the analogous electric quadrupole. Is this compatible with the claim that the gravitomagnetic field is completely analogous to electromagnetism?
On December 29, 2021 we tried to calculate the energy content of a gravitational wave by considering its positive and negative pressure effects on a solid body. Our analysis suggested that the large energy density of a gravitational wave comes from its ability to stretch spatial distances.
This is a property which does not exist in an electromagnetic wave. In this respect gravity is not analogous to electromagnetism. If that is the case, can we define magnetic gravity in a reasonable way at all?
Harvesting energy from a wave
If we have any kind of a wave, we can harvest energy from it by putting a system of charges, let us call that S, into it and letting it move those charges. If we put a force which makes the charges resist the movement, we can harvest energy. The movement of the charges must in this case "cancel" part of the wave. By calculating the cancellation effect we can deduce the energy density of the wave.
However, the cancellation effect is not straightforward to determine because, e.g., the electromagnetic wave is not "linear" on the charges generating it. Let D be a dipole antenna generating a wave. If we let another dipole antenna D' harvest energy from the wave, then D' can oscillate almost in sync with the incoming wave, but D' does not contribute to the amplitude of the wave – on the contrary.
Let us consider mechanical waves in a tense string. We can "couple" masses m to the string with springs. The interaction force F between the wave and the mass m does not reveal much about the string.
Conclusions
The energy density of a gravitational wave is a very important concept if we want to understand magnetic gravity. Let us write a new blog post about it.
Stephen M. Barnett (2013) studies the analogy between gravitational waves and electromagnetic waves.
In the link, Chris Hirata (?) (2018) derives the energy density by first forming a wave solution in linearized Einstein equations, and then calculating the error term that the wave solution has in full, nonlinear, Einstein equations. The error term can be interpreted as a "stress-energy tensor" T. The mass-energy density T₀₀ is then taken as the energy density of the gravitational wave.
Why does this work?
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