Possible changes in the Law of Gravitation

[mishrashubham]

Now there's this paragraph in my physics textbook that just completely went over my head.

There is some indication that the force between two masses is not as described in this chapter. The deviation from the simple law F=GMm/r^2 is being taken as an indication of the existence of a fifth interaction besides gravitational, electromagnetic, nuclear and weak. It has been reported (Phys. Rev. Lett. Jan 6 1986) that the force between two masses may be better represented by

$$F=\frac{G_\infty m_1 m_2}{r^2}\bigg[1+\bigg(1+\frac{r}{\lambda}\bigg)\alpha e^{\frac{-r}{\lambda}}\bigg]$$

with $$\alpha\approx -0.007$$ and $$\lambda\approx 200 m$$. As $$\alpha$$ is negative, the second term in the square bracket represents a repulsive force. For r>>200 m

$$F=\frac{G_\infty m_1 m_2}{r^2}$$

which is the force operative between the Earth and other objects. For r<<200 m

$$F=\frac{G_\infty m_1 m_2(1+\alpha)}{r^2}=\frac{G' m_1 m_2}{r^2}$$

where $$G'=G_\infty(1+\alpha)$$

This is the force we measure in a Cavendish-experiment. The value of G for small distances is about 1% less than the value of G for large distances.

What does this mean? I could not understand a single word.

 

[superg33k]

Interesting! Whats the name of the book?

 

[jtbell]

And how old is it? The experiment or analysis which claimed to have found evidence for a "fifth force" was twenty years ago, or more, and has since been refuted IIRC.

Or maybe this is about a different "fifth force" than the one I'm thinking of, which was proposed by a guy names Fischbach, I think.

 

[mishrashubham]

Interesting! Whats the name of the book?

The name of the book is "Concepts of Physics by HC Verma" (An Indian Author).

And how old is it? The experiment or analysis which claimed to have found evidence for a "fifth force" was twenty years ago, or more, and has since been refuted IIRC.

Or maybe this is about a different "fifth force" than the one I'm thinking of, which was proposed by a guy names Fischbach, I think.

There is only a single edition (unfortunately) and it came out in 1992.

Edit : BTW, the mathjax testing is making it look a bit weird. However the zoom function is cool.

 

[Rear Naked]

why is the lambda = 200m, and what is 200m the lambda of?

 

[mishrashubham]

why is the lambda = 200m, and what is 200m the lambda of?

Exactly, I don't know! Neither do I know what alpha is nor do I know what e is. I posted here so that people who who are familiar with this equation or have used this equation somewhere, can help me understand what it means and what it is trying talk about.

 

[mishrashubham]

Or maybe this is about a different "fifth force" than the one I'm thinking of, which was proposed by a guy names Fischbach, I think.

Thanks for the name. The wikipedia page says

On April 6, 2011 there was a possible discovery at Fermilab’s Tevatron that, according to physicists, could transform all of high energy physics. The possibility of discovering the elusive 'fifth force' is based on 4.3 inverse femtobarns of data. The CDF collaboration will repeat the analysis with at least twice as much data to see whether the bump gets more or less pronounced. Other experiments, including DZero and the LHC experiments, will look for a particle of about 140 GeV/c2 in their data as well.

and also provides a reference. http://www.symmetrymagazine.org/breaking/2011/04/07/fermilabs-data-peak-that-causes-excitement/

http://en.wikipedia.org/wiki/Fifth_force

 

[Rear Naked]

maybe Fermilab won't be pwnt by the LHC after all

 

[mishrashubham]

maybe Fermilab won't be pwnt by the LHC after all

That is not the point.

Anyways; has anyone ever seen this equation?

 

[diazona]

I haven't seen this particular equation, probably because according to the Wikipedia article, the results have not been duplicated (making them effectively discredited).

But anyway, it's not that hard to understand, especially if you go back to the original paper from 1986. Basically, the authors of that paper (Fischbach et al.) were looking at some experimental measurements of the gravitational constant, $G$. They noticed that geophysical measurements of $G$, i.e. measured using variations in gravity around the Earth, were giving slightly larger numbers than the "laboratory value" of $G$ which is based on small-scale measurements using an Eötvös torsion balance and the like. The point is that the torsion balance measurements involved objects which were not as far apart as the objects in the geophysical measurements, e.g. two spheres in a lab are not as far apart as the Earth and a satellite, so if the force between two massive objects depends on their separation in some way other than $F \propto 1/r^2$, it would show up as a discrepancy in the value of $G$ between these two different classes of experiments.

Now, the normal gravitational potential energy of two masses is given by
$$V_g(r) = -\frac{Gm_1 m_2}{r}$$
If there is another force at work, because of calculations from quantum field theory, we would expect it to make an additional contribution to the potential energy of the form
$$V_5(r) = -Gm_1 m_2\frac{\alpha e^{-r/\lambda}}{r}$$
The constant $\lambda$ can be loosely interpreted as the "range" of the force, and $\alpha$ is related to the strength of the force. So the total potential, corresponding to gravity plus this "fifth force," would be
$$V(r) = -\frac{Gm_1 m_2}{r}(1 + \alpha e^{-r\lambda})$$
The authors of the 1986 paper checked the experimental data to see whether it was consistent with a potential energy of this form, and they concluded that it was. By performing a fit to the data (I presume) they identified the specific values of $\alpha$ and $\lambda$ that best matched the measurements, namely $\alpha = (-7.2\pm 3.6)\times 10^{-3}$ and $\lambda = (200 \pm 50)\ \mathrm{m}$.

The excerpt from the textbook talks about the force that would correspond to this potential energy, which you can derive using the usual definition $F = -\partial V/\partial r$. Basically it is showing you mathematically that, at large distances (much larger than the "range" $\lambda$), the "fifth force" is negligible, so if you measure the force between masses at large distances (e.g. with satellites and the Earth) it will look just like the normal gravitational force. But at smaller distances, e.g. within a lab, the "fifth force" is not negligible, and in fact the way that it manifests itself is as an extra contribution with a strength $\alpha$ times the gravitational force. You can see this by doing a Taylor expansion in $r/\lambda$. This explains why, if you measure the gravitational constant in a lab assuming that the only force involved is ordinary Newtonian gravity, you will get a value that differs from larger-scale measurements by a factor of $1 + \alpha$.

 

[mishrashubham]

Thank you
I am not sure I followed the calculations from quantum field theory (yukawa potential), but I got the basic idea so thanks again.

 

[diazona]

Thank you
I am not sure I followed the calculations from quantum field theory (yukawa potential), but I got the basic idea so thanks again.

Unless you've studied enough QFT that you should know how to calculate the amplitude corresponding to a Feynman diagram, I wouldn't worry about it.

 

[mishrashubham]

Unless you've studied enough QFT that you should know how to calculate the amplitude corresponding to a Feynman diagram, I wouldn't worry about it.

Well I haven't studied any QFT so I think that would be fine.