# Could a force with these properties exist in a parallel universe?

In an alternate universe there is a force known as the emotion force although the force has nothing to do with emotions. Anything that interacts through the emotion force has an emotion charge that is either positive or negative. Emotion charge is entirely independent of electric charge. Like emotion charges attract while opposite charges repel. The force between two emotion charges is equal to the square root of the product of their emotion charges divided by the distance between them squared multiplied by a constant. The force carrier for the emotion charge is massless and so the emotion force has an infinite range. The strength of the emotion force is about 1/100,000.

Could a universe have a force like this and what effects would this kind of force have on physics?

• You haven't really explained it that well. What produces this force? Does everything inherently have this force? are you just asking if an electric charge had slightly different properties? – Aequitas Oct 21 '15 at 2:20
• You can make any laws of physics you want, if you create a universe. You can create new particles and fields, and thus new forces. There's nothing stopping you. All you've done is create a force that behaves exactly like the electric force, if I'm not mistaken, in which case the answer is yes. Finally, how are you measuring "strength"? What do you mean by "1/100,000"? – HDE 226868 Oct 21 '15 at 2:25
• The force between opposite emotion charges is complex. That's a problem. – Alex S Oct 21 '15 at 2:28
• @Aequitas, The force would be produced by particles that would have emotion charge that would be an intrinsic property of the particles like how electric charge is an intrinsic property in our universe. Not everything would have this force, just things that would have emotion charge and not everything would have emotion charge. It would exist in addition to electric charge but would not replace electric charge. – Anders Gustafson Oct 21 '15 at 2:29
• I seem to recall we had a similar question not that long ago. One thing you'd have to work out is whether the force is in any way related to the size, mass or something else of the objects involved. (Like the force of gravity at equal distance being a function of the masses involved.) If it is related to size/mass/etc. of the objects involved, you need something that limits the force or you might well end up with a chain reaction that results in your universe containing two unimaginably huge objects and nothing else. At that point, because of gravity, you'd likely be looking at singularities. – user Oct 21 '15 at 8:30

Yes.

We can write out the force as $$\mathbf{F}=-k\frac{\sqrt{e_1e_2}}{r^2}\mathbf{u}_r$$ where $\mathbf{u}_r$ is the unit vector in the direction of $\mathbf{r}$, defined as $$\mathbf{u}_r=\frac{\mathbf{r}}{r}$$ Is this possible? Well, it's an inverse-square law (i.e. $\mathbf{F}\propto r^{-2}\mathbf{u}_r$), and we have two of those in our universe (gravity and the electric force, at least in classical approximations). The only problem is, as Alex mention, the square root. If $\text{sgn}(e_1)=-\text{sgn}(e_2)$, then we have the square root of an imaginary number, and thus a vector of imaginary magnitude, which is not a good thing.

This can be fixed, though. Simply modify the equation to be $$\mathbf{F}=-k\frac{\sqrt{|e_1e_2|}\text{sgn}(e_1)\text{sgn}(e_2)}{r^2}\mathbf{u}_r$$ Here, $\text{sgn}(x)$ is the sign function, defined as $$\text{sgn}(x)=\frac{|x|}{x}=\frac{x}{|x|}$$ This gives us a force that is proportional to the square root of the product of the magnitudes of the charges, as intended, while preserving the property that opposites repel, while like charges attract.

On a different note, it's your universe. In many cases, you can do whatever the heck you want. There are some cases where this doesn't hold, but this isn't one of those exceptions.

• However, if we divide $e_1$ into two parts, the force will rise $\sqrt{2}$ times, from $k \sqrt{e_1 e_2}/r^2$ to $2 k \sqrt{(e_1/2) e_2}/r^2$. What to do with this? If the force acts only between elementary particles, either there exist only one value of emotion charge and the square root is not important (maybe the effective charges are, eg. $1$&$\sqrt{2}$ - it is possible, it is a problem, why electric charges are rational), or the effective charge (coupling to the carrier of this force) depends somehow on transformation between one charged particle and virtual pair of two charged particles. – BartekChom Oct 21 '15 at 6:24
• I think you mean if $sgn(e_1)=-sgn(e_2)$. The magnitudes will never have opposite signs. @BartekChom, I'm not sure where you're getting a factor of $\sqrt{2}$. The sign function always returns $1$ or $-1$. The only difference between the first and second functions is the first might produce $\sqrt{x}i$ while the second would produce $-\sqrt{x}$ instead. – MichaelS Oct 21 '15 at 7:11
• @MichaelS: The force between one charge $e_1$ and one charge $e_2$ is $k \sqrt{e_1 e_2}/r^2$, but the force between two charges $e_1/2$ and one charge $e_2$ is $2 k \sqrt{(e_1/2) e_2}/r^2=\sqrt{2} k \sqrt{e_1 e_2}/r^2$. I am not talking about signs. I am sorry, did I write something unclear? – BartekChom Oct 21 '15 at 16:36
• @MichaelS Yes, ou're right. – HDE 226868 Oct 21 '15 at 22:51
• @BartekChom What? No. All I'm doing is taking the absolute value of $e_1e_2$, which is the quantity mentioned in the question. I'm not dividing either $e$ by anything. – HDE 226868 Oct 21 '15 at 22:52

Note that we can actually reformulate the "emotion force" to discard the square roots. I'll call my formulation the clumping force (for reasons that will become clear later).

First I define a clumping charge $\chi$ in terms of your "emotion charge" $e$:

$$\chi = \text{sign}(e)\sqrt{|e|}$$

Then we can write the (attractive) clumping force between two particles (equal to the "emotion" force) to be:

$$F_e = F_\chi \\ k_e\frac{\text{sign}(e_1)\text{sign}(e_2)\sqrt{|e_1| |e_2|}}{r^2} = k_\chi\frac{\chi_1\chi_2}{r^2}$$

We can see that the clumping force (and therefore the "emotion" force) is an ordinary inverse-square force, and we can use all the tools we've developed for similar forces like gravity and the electrostatic force.

I just want to point out that the clumping force is the "emotion" force, just with a more mathematically convenient way of expressing the charge. If you're really set on this square-root thing, go ahead, but just remember that the "emotion charge" is not additive (i.e. $e_\text{tot} = (\sqrt{e_1}+\sqrt{e_2})^2 \neq e_1 + e_2$). (By the way, if this force has "nothing to do with emotions" I would suggest picking a different name.)

Now the effect of a force depends not only on its strength, but also how its charge is distributed through matter. For example, gravity is a stunningly weak force, but totally dominates the behavior of matter on large scales. In fact, in our everyday lives the electromagnetic force only becomes dominant at microscopic scales.

This is due to the fact that most objects are almost exactly electrically neutral: they carry no net electrical charge. On the other hand, every object has nonzero gravitational charge, and the mass of the Earth is huge, so the "weak" gravitational force is relatively strong on large scales.

Why is there such an imbalance? The answer is the "opposites attract" behavior of the electric force. If you try to assemble two charged particles into a more charged object, the particles will naturally repel and dissipate. On the other hand, if you try to assemble a neutral object from two oppositely charged particles, the electric force assists you and the object tends to stay together. This results in a natural evening-out of the electric charge.

Gravity, on the other hand, is always attractive, and therefore tends to assemble particles into larger objects with more mass. This results in the distribution we see today, with a small number of very massive objects surrounded by almost-empty space.

The emotion/clumping force is like gravity: since like charges attract, charged particles will tend to assemble into larger, even more charged objects. (Now you see the reason for the name.) As Michael Kjörling posits this could very well result in the eventual segregation of of all particles into two oppositely-charged clumps speeding away from each other. Every particle not part of one of the clumps will accelerate towards the clump of the same charge and away from the clump of the opposite charge.

Gravity would seem to act in the same way (except that only one clump would form, since there is no negative mass), and indeed it does tend to clump matter together. The Universe temporarily avoids this fate because the strong force is so much stronger than gravity: it tends to result in the explosion of clumps that grow too massive.

Thus you basically have three options for this force:

• If the "emotion force" is stronger than the strong force, this will lead to the fate I described above (collapse of all "emotionally charged" matter into two oppositely-charged black holes).
• If it is weaker than the strong force but stronger than gravity, then the universe will consist of "emotionally-bound" stars, which will be smaller and burn much faster than the gravitationally-bound stars of this universe.
• If it is weaker than gravity, then you will get gravitationally-bound objects that tend to segregate themselves into two oppositely-charged halves. Depending on what types of particles have "emotion charge," this segregation could be impeded or even eliminated by the formation of neutrally-charged atoms or molecules.

tl;dr: No, it is not possible to have the force proportional to the product of the square roots of the charges.

In an alternate universe there is a force known as the emotion force although the force has nothing to do with emotions. Anything that interacts through the emotion force has an emotion charge that is either positive or negative. Emotion charge is entirely independent of electric charge.

So far, no problem. There's nothing wrong with having different types of charges which are independent of each other. Indeed, already in our world, we have that.

Like emotion charges attract while opposite charges repel.

I have a feeling that this already should give problems, but I can't pinpoint it, so possibly that's no problem either.

The force between two emotion charges is equal to the square root of the product of their emotion charges divided by the distance between them squared multiplied by a constant.

Here is the problem. While the inverse square of the distance is good (indeed, for a long-range force it's the only reasonable choice), the square root of the emotion charge is not.

Consider the following:

You have two objects of equal emotion charge close to each other, and a third one far enough away that you can in good approximation consider them to be at the same place. Let's call the charge of the far-away particle $q_0$, and the charges of the other two particles $q_1$ and $q_2$

Then you can calculate the force on the far-away particle in two different ways:

1. The force is the sum of the forces due to the attraction from each individual object: $$F = C \frac{\sqrt{q_0}\sqrt{q_1}}{r^2} + C \frac{\sqrt{q_0}\sqrt{q_2}}{r^2} = C\frac{\sqrt{q_0}}{r^2}\left(\sqrt{q_1}+\sqrt{q_2}\right)$$

2. The close-together individual objects are considered a combined object, whose charge is then of course $q_1+q_2$. The force between the far-away object and the combined force is therefore $$F = C\frac{\sqrt{q_0}\sqrt{q_1+q_2}}{r^2} = C\frac{\sqrt{q_0}}{r^2}\left(\sqrt{q_1+q_2}\right)$$

Clearly it is not possible for both equations to be fulfilled at the same time, unless one of the charges is zero.

The force carrier for the emotion charge is massless and so the emotion force has an infinite range. The strength of the emotion force is about 1/100,000.

No problems with this (indeed, a massless force carrier is both necessary and sufficient for an $1/r^2$ force).