I know that the wave lengths would be shorter but how would this extend habitable zones? Would zones disappear? Would other stars have an effect on Earth? If so, then would that counteract the ability to traverse space because you would have too distant stars?
I'm using this for my book. I wanted it to be as realistic as possible but by changing this law only. The relative speeds of everything else is about the same.
The question is quite broad so I will only focus on the habitable zones problem you also asked.
$$4 \ _1^1H \rightarrow \ ^4He^{2-} + 2e^+ + 2v_e$$
That is one of the most common fusions, the proton-proton chain reaction. It isn't exactly one fusion but a set of fusions which produces a helium atom as a final product.
This is the mass of hydrogen-1:
$$_1^1H = 1.00782503224194 \ u$$
This is the mass of the positron (e+):
$$e^+ = 5.48579909016\times10^{−4} \ u$$
And this is the mass of helium:
$$_2^4He = 4.002602 \ u$$
$v_e$ doesn't have mass since its energy:
Now solve the above equation using the mass of each particle:
$$4 \times 1.00782503224194 = 4.002602 + 5.48579909016\times^{10−4} + 2v_e $$ $$4.03130012896776 = 4,003150579909016 + 2v_e$$ $$4.03130012896776 - 4,003150579909016 = 2v_e$$ $$0,028149549058744 \ u = 2v_e$$
As you can see there is a net loss of mass that is transformed into energy. For being exacted it's a $0,00698269267408968993978186921633\%$, or $0.007\%$ for short.
That $0.007\%$ is the famous rate of mass to energy turned on the H->He fusion.
Now, what I say all that?
Well, mass-energy equivalence. That famous equation is:
$$E = mc^2$$
As you can read above $0,028149549058744 \ u$ of mass is transformed into energy, obviously using the above formula. So if you increase by ten the speed of light ($c$), the amount of energy produced (due mass to energy conversion) will also increase at least by ten (remember, squared $c$).
So, in addition to all the problems, you will find increasing the speed of light, like the change of wavelength or chemical bonds, your stars will output a hundred times (due $c^2$) more energy/light/heat, I am quite sure that won't be good for habitable zones.
I know that the wave lengths would be shorter
Wrong, this is backwards. Wavelength times frequency equals speed, and with speed increasing by a factor of ten, so would the other half of the equation. Since energy scales with frequency, the same energy photons would have longer wavelengths. This means that anything that relies on electromagnetic radiation would need to be ten times as large to work. Antennas might be too small to actually receive radiation, and anything with lenses would now be blurry.
habitable zones would zones disappear? Would other stars have an effect on earth?
No and no. The inverse-square law is a matter of geometry and energy. Nothing would change. The sun would still produce the same amount of energy and the Earth is still the same distance away, so it's fine.
Although you should see below for an actual answer.
I wanted it to be as realistic as possible but by changing this law only
I would advise not doing that.
Look, the speed of light is heavily bound in up in several fundamental universal constants. And screwing with those is a bad idea. You're absolutely going to bring the electromagnetic force out of balance with the other forces, which is going to have interesting consequences for anything that this happens to.
I don't know enough physics to describe all the consequences, but nothing is going to survive this field. No person, place, or plot device will survive this. Planets would only "survive" by shear virtue of their mass meaning that when the change reverts, they fall back together and there's still a pile of mass of about the same size in the same location.
For instance, the charge of an electron, e:
$$ e={\sqrt {\frac {2\alpha h}{\mu _{0}c_{0}}}}={\sqrt {2\alpha h\varepsilon _{0}c_{0}}}. $$
(From Wikipedia)
That c0 is the speed of light, and the elementary charge changing to be more than three times its normal value means that every single molecular bond in the area of the effect stops being a thing. It probably also means that stars stop working, because the increased strength of the electromagnetic force means that the stars no longer have enough pressure to overcome the repulsion of the nuclei to cause fusion.
So the habitable zones? They're actually way shrunk in now, because stars have to be massive to even have a trickle of fusion going on.
