# Bohr Radius of Uniformly Accelerating Hydrogen Atom

Suppose we have a rocket-ship that is uniformly accelerating along its bow-stern axis (connecting line). Also, assume we have a hydrogen atom in the rocket.

Mathematically (theoretically), does general relativity predict that the Bohr radius of that hydrogen atom could be different at the bow than at the stern? This is a specific example of my general question of can uniform acceleration (as in this simple rocket-ship example) have an impact on the "shape" or "size" of an atom, in some way.

Relativistic treatment for Bohr's atom is explored in this paper by A.F. Terzis. It presents a concise derivation of the predictions of Bohr’s theory, describing the dynamics of the atoms using special relativity. Though this does not essentially invoke the principles of general relativity, I believe one could gain some insight for understanding the effects of relativity on atomic structure.

Results: For the hydrogen atom, it is shown that for most practical cases the main results do not differ appreciably from the findings of the classical model. In contrast, it is revealed that for hydrogen-like atoms with heavy nuclei, the predictions of the relativistic model differ significantly compared to the prediction of the classical Bohr model.

• The radius of each orbit for Hydrogen atom is found out to be $$r_n = \frac{\hbar^2}{me^2}n\sqrt{n^2-\alpha^2} = a_0n\sqrt{n^2-\alpha^2}$$ with $$\alpha$$ being the fine structure constant and $$a_0 \equiv \dfrac{\hbar^2}{me^2} = 0.529 {\mathring{\text{A}}}$$.

• However, for the radius for hydrogenic atoms with the atomic number $$Z$$, a surprising result was obtained. $$r_n = \frac{a_0n\sqrt{n^2-Z^2\alpha^2}}{Z}$$

It is easy to see that for $$n=1$$ and $$Z>1/\alpha$$, we will have an imaginary radius and hence forbids the existence of elements with those atomic numbers. $$1/\alpha$$ serves as an upper limit.

Reference: A simple relativistic Bohr atom, Terzis A F, Eur. J. Phys. 29 (2008) 735–743,doi:10.1088/0143-0807/29/4/008

• Thank you, I will take a look. I am just not sure if space, which I understand is dilated at the bow compared to at the stern, theoretically could change the bohr radius of hydrogen (even a little) or if the electromagnetic forces of the atom maintain it. – David Oct 8 '19 at 17:10