# Potential energy and frame of reference

I was studying about potential energy, and I suddenly thought that is there any relation between potential energy and frame of reference?

For example, we say for an object raised to a height $h$, potential energy is $mgh$. But this is so when we are talking about its distance from Earth. If a person is holding a suitcase then for the person shouldn't be the P.E. of suitcase be zero?

The potential energy has a gauge freedom, that is we can define the zero to be anywhere we want without affecting the physics. A side effect of this is that we cannot experimentally measure potential energy, we can only measure differences in potential energy.

So when you say the potential energy of an object raised to a height $h$ is $mgh$, what this really means is that raising an object by a distance $h$ changes the potential energy by $mgh$ i.e. the difference in the potential energy before and after raising was $mgh$.

The person holding the suitcase can define its potential energy to be zero, but this is just a choice of gauge. Regardless of how the person holding the suitcase defines the potential energy it still changes by $+mgh$ when it is raised by a distance $h$ and $-mgh$ if it is lowered by a distance $h$.

• what do you mean by gauge?? – Vidyanshu Mishra Sep 9 '16 at 16:50
• @VidyanshuMishra: in general gauge freedom is quite a complicated idea. However in this case it just means we are free to set the zero of the potential energy anywhere we want. – John Rennie Sep 9 '16 at 17:00
• means freedom to choose the origin?? – Vidyanshu Mishra Sep 9 '16 at 17:02
• and what does that star mean??(on the left side of question) – Vidyanshu Mishra Sep 9 '16 at 17:07
• @VidyanshuMishra: yes, in this case the gauge freedom is the freedom to choose the origin. If you click on the star next to the question it adds the question to your favourites list. – John Rennie Sep 9 '16 at 17:11

One important point which is often missed it is the not the object alone that has the gravitational potential energy.
It is the object and the Earth which has the gravitational potential energy.
A change in the gravitational potential energy occurs when the separation of the object and the Earth changes.
When you raise the object by a distance $h$ the gravitational potential energy increases by $mgh$.
So whether you are holding the object or not the change in gravitational potential energy is still $mgh$ when the object is raised by a distance $h$.

It is often convenient to define the gravitational potential energy as zero when the separation of the object and the Earth is a certain value.
So it might be when the object is on the surface of the Earth or on a laboratory bench.
So instead of repeatedly stating that there is a change in the gravitational potential it is more convenient to state that the gravitational potential is $mgh$ where $h$ is the height of the object above the surface of the Earth or the laboratory bench.

The $h$ in the formula $$U=mgh$$ is the vertical distance between two points. And those two points can be chosen freely - it's not about Earth.