Gravitomagnetic Experiment: Possibility & Calculations

[olgerm]

[Mentor Note -- LaTeX edited for readability]

I was thinking about an experiment to demonstrate gravitomagnetic effect. I did my calculations using gravitomagnetic model. It is not as accurate as general relativity, but GR should give similar predictions. I do not know if it would be possible to to this experiment in real life(are there enough accurate sensors and tough materials).
Installations consists of:

• a spinning cylinder with hole in it, that is for creating gravitomagnetic field. last one is for detecting gravitomagnetic field. axis of cylinder is parallel to coorinate-z-axis.
• laser, that sends it's ray throght the hole in cylinder.Ray is directed to z axis, and is parallel to y-axis. x-coordinate of laser location is 0.
• detector, that detect angel of the laserray. (may be something that's measures location of laserray after it has traveled a long distance)

I make an approximation, that the hole is much smaller than the cylinder and ignore it. Gravitimagnetic field created by the spinning cylinder (look derivation from https://www.physicsforums.com/threads/gravitomagnetic-experiment.824048/):
$\vec B(x, y, R, H, H_0)= (\frac{k_G4πρω}{c^2}\int_{H_0}^{H+H_0}(\int_0^R((\frac{hπrx(\sqrt{h^4(x^2+y^2-r^2)^2+2h^2(r^2+x^2+y^2)}-(h^2+r^2+x^2+y^2))}{(x^2+y^2)\sqrt{h^4+(x^2+y^2-r^2)^2+2h^2(r^2+x^2+y^2)}})dr)dh),\frac{k_G4πρω}{c^2}\int_{H_0}^{H+H_0}(\int_0^R((\frac{hπry(\sqrt{h^4+(x^2+y^2-r^2)^2+2h^2(r^2+x^2+y^2)}-(h^2+r^2+x^2+y^2))}{(x^2+y^2)\sqrt{h^4 + (x^2+y^2-r^2)^2+2h^2(r^2+x^2+y^2)}})dr)dh),\frac{k_G4πρω}{c^2}\int_{H_0}^{H+H_0} (\int_0^R (\int_0^{2π} (\frac{(r-xCos(a) - ySin(a))r^2}{(x^2+y^2+r^2+h^2-2r(xCos(a)+ySin(a)))^{3/2}} dα)dr)dh))$
since unlike in this post cylinder is for simplicity assumed to be infinitly long:
$\vec B(x, y)= (\frac{k_G4πρω}{c^2}\int_{-\infty}^{\infty}(\int_0^R((\frac{hπrx(\sqrt{h^4(x^2+y^2-r^2)^2+2h^2(r^2+x^2+y^2)}-(h^2+r^2+x^2+y^2))}{(x^2+y^2)\sqrt{h^4+(x^2+y^2-r^2)^2+2h^2(r^2+x^2+y^2)}})dr)dh),\frac{k_G4πρω}{c^2}\int_{-\infty}^{\infty}(\int_0^R((\frac{hπry(\sqrt{h^4+(x^2+y^2-r^2)^2+2h^2(r^2+x^2+y^2)}-(h^2+r^2+x^2+y^2))}{(x^2+y^2)\sqrt{h^4 + (x^2+y^2-r^2)^2+2h^2(r^2+x^2+y^2)}})dr)dh),\frac{k_G4πρω}{c^2}\int_{-\infty}^{\infty} (\int_0^R (\int_0^{2π} (\frac{(r-xCos(a) - ySin(a))r^2}{(x^2+y^2+r^2+h^2-2r(xCos(a)+ySin(a)))^{3/2}} dα)dr)dh))$

• $k_G$ is gravitational constant.
• $\rho$ is density of cylinder.
• $R$ is radius of the cylinder.

force on a single photon is:
$\vec{F}=\vec{v}\times\vec{B}m_{photon}4$

as approximation I can calculate force as:
$\vec{F}\approx (cB_zm_{photon}4,0,0)$

acceleration of photon is:
$\vec{a}=\frac{\partial \vec{X}^2}{\partial t^2}=(\frac{F}{m_{photon}}-\frac{(\vec F.\vec v)\vec v}{m_{photon}c^2})\frac{\sqrt{c^2-v^2}}{c}$

as approximation I can calculate acceleration of photon as:
$\vec{a}\approx\frac{\partial \vec{X}^2}{\partial t^2}=\frac{F}{m_{photon}}$

speed is:
$v=\int(dt\vec{a}(t))$

change of direction of laserray is:
$\alpha=arcsin(\frac{\sqrt{v_x^2+v_z^2}}{c})$

as approximation I can calculate change of direction of laserray as:
$\alpha\approx arcsin(\frac{v_x}{c})$

by simplifying equations above I get that approximate change of direction of laserray is:
$\vec{a}\approx (cB_z4,0,0)$
$\vec{v}=\int(dt\vec{a}(t))\approx\int(dt\vec{(cB_z(\vec{x}(t))4,0,0)}(t))$
I make an approximation, that gravitomagnetic fileld on path of photon is same as in path that photon would take if it traveled straight.:
$\vec{v}=\int(dt\vec{a}(t))=\int(dt\vec{(cB_z(\vec{x}(t))4,0,0)}(t))=\int(dl\vec{(B_z(\vec{x}(t))4,0,0)}(l))$
$\alpha\approx arcsin(\frac{\int(dl\vec{(B_z(\vec{x}(t))4)}(t))}{c})=arcsin(\frac{k_G16πρω}{c^3}\int(dl\vec{(\int_{-\infty}^{\infty} (\int_0^R (\int_0^{2π} (\frac{(r-xCos(a) - ySin(a))r^2}{(x^2+y^2+r^2+h^2-2r(xCos(a)+ySin(a)))^{3/2}}dα)dr)dh))})) =arcsin(\frac{k_G16πρω}{c^3}\int(dl\vec{(\int_{-\infty}^{\infty} (\int_0^R (\int_0^{2π} (\frac{(r-ySin(a))r^2}{(y^2+r^2+h^2-2r(ySin(a)))^{3/2}}dα)dr)dh))}))$

($m_{photon}$ is 0 . It cancels out. It is in formulas as dummy-variable))

Maybe adding some mirrors somewhere would make the experiment even better.

Are these equations correct?
Would it be possible to make such experiment?
I hope someone takes some time and writes clear arguments why such experiment would be or would not be possible or writes some ideas how to change this experiment.

 

[weirdoguy]

Are these equations correct?

They are as always unreadable. You've been told repeatedly that $*$ is not a symbol for multiplication, and there is no need to add it zilions of times.

 

[berkeman]

They are as always unreadable. You've been told repeatedly that $*$ is not a symbol for multiplication, and there is no need to add it zilions of times.

I tried my best to delete the "*" characters in the LaTeX. Let me know if anything looks wrong still.

@olgerm -- Please do not use the "*" character for multiplication in LaTeX. It is a symbol used for convolution, not multiplication in math typesetting. You've never seen it used in textbooks for multplication, so please try to use LaTeX here to create textbook-style math equations. Thank you.

 

[PeterDonis]

I do not know if it would be possible to to this experiment in real life

As you have already been told, the gravitomagnetic effects of objects of ordinary size are many, many orders of magnitude too small for us to detect with any technology we have now or will have in the foreseeable future. You need rotating masses of planetary size at least, and even then it takes very precise measurements that are just at the edge of our current capabilities (such as Gravity Probe B detecting the Lense-Thirring precession due to the Earth).

 

[olgerm]

As you have already been told, the gravitomagnetic effects of objects of ordinary size are many, many orders of magnitude too small for us to detect with any technology we have now or will have in the foreseeable future. You need rotating masses of planetary size at least, and even then it takes very precise measurements that are just at the edge of our current capabilities (such as Gravity Probe B detecting the Lense-Thirring precession due to the Earth).

Centre of the artificial cylinder could be much closer to sensor than center of Earth could be. And it's angular speed could be bigger. I am not sure if this experiment would be possible, but I need more clear explanation(How big could the $B_G$ field be; how much it would change angle of laserray) to be conviced that it is not possible.

 

[PeterDonis]

Centre of the artificial cylinder could be much closer to sensor than center of Earth could be. And it's angular speed could be bigger.

Doesn't help. You're still many, many orders of magnitude away from having an effect detectable with our current or foreseeable future technology.

I am not sure if this experiment would be possible, but I need more clear explanation(How big could the $B_G$ field be; how much it would change angle of laserray) to be conviced that it is not possible.

No, you don't need "more clear explanation". I've already given you the basic explanation. You need to do some actual calculations: plug numbers into your equations and see what they tell you. What they will tell you is that my statement, that the effect in this experiment is many, many orders of magnitude away from being detectable with our current or foreseeable future technology, is correct.

 

[Vanadium 50]

Centre of the artificial cylinder could be much closer to sensor than center of Earth could be. And it's angular speed could be bigger

Woulda, shoulda, coulda.

I am not sure if this experiment would be possible,

Then why are you arguing with us? You need planetary masses to see any effect. You have two choices - believe us, or show us quantitatively that you are right and we are wrong. A page of unreadable LaTeX that never reaches a number doesn't count - it just wastes everybody's time.

 

[olgerm]

Woulda, shoulda, coulda.

Reason I wrote that was to say that just mass of planets being bigger and (Newtonian)gravitational field of planets being bigger than lab objects does not make it obvious that gravitomagnetic field of planets is bigger than gravitomagnetic field of lab objects.
I have done quite a lot work and written quite clearly my idea and formulas (without numeric claclulations). I expect clear explanation WHY this experiment would be impossible. Just to say that it is impossible that gravitomagnetic field of Earth is hardly measurable and lab objects are smaller than Earth is no sufficient explanation (read above why is not sufficent).

 

[PeterDonis]

I expect clear explanation WHY this experiment would be impossible.

Sorry, if what you've already been told is not enough, we can't help you any further. Go do the math yourself and see what it tells you. You've already been told what it will tell you, but if you're not going to believe what you've been told, you need to go calculate the answer for yourself. Further discussion here is pointless.

Thread closed.