Let's say there are 2 people, A and B. They both are at the sea level. A gets higher than B. Does time moves faster to A then B? (Does height -in other words gravity- causes Time Dilation?)
General Relativity – Does Gravity Cause Time Dilation?
general-relativitygravityspecial-relativitytimetime-dilation
Related Solutions
The time dilation is due to a difference in the gravitational potential energy, so it is due to the difference in height. It doesn't matter whether the strength of the gravitational field varies, or how much it varies, all that matters is that the two observers comparing their clocks have a different gravitational potential energy.
To be more precise about this, when the gravitational fields are relatively weak (which basically means everywhere well away from a black hole) we can use an approximation to general relativity called the weak field limit. In this case the relative time dilation of two observers $A$ and $B$ is given by:
$$ \frac{dt_A}{dt_B} \approx \sqrt{ 1 + \frac{2\Delta\phi_{AB}}{c^2}} \approx 1 + \frac{\Delta\phi_{AB}}{c^2} \tag{1}$$
where $\Delta\phi_{AB}$ is the difference in the gravitational potential energy per unit mass between $A$ and $B$.
Suppose the distance in height between the two observers is $h$, then in a constant gravitatioinal field with acceleration $g$ we'd have:
$$ \Delta\phi_{AB} = gh $$
If this was on the Earth then taking into account the change in the gravitational potential energy with height we'd have:
$$ \Delta\phi_{AB} = \frac{GM}{r_A} - \frac{GM}{r_B} $$
where $r_A$ and $r_B$ are the distances of $A$ and $B$ from the centre of the earth and $M$ is the mass of the Earth. Either way when we substitute our value of $\Delta\phi_{AB}$ into equation (1) we're going to get a time dilation.
As for the accelerating rocket: the shortcut is to appeal to the equivalence principle. If acceleration is equivalent to a gravitational field then it must also cause a time dilation in the same way that a gravitational field does.
Alternatively we can do the calculation rigorously. The spacetime geometry of an accelerating frame is described by the Rindler metric, and we can use this to calculate the time dilation. The Rindler metric for an acceleration $g$ in the $x$ direction is:
$$ c^2d\tau^2 = \left(1 + \frac{gx}{c^2}\right)^2c^2dt^2 - dx^2 - dy^2 - dz^2 $$
We get the time dilation by setting $dx=dy=dz=0$ to give:
$$ c^2d\tau^2 = \left(1 + \frac{gx}{c^2}\right)^2c^2dt^2 $$
and on rearranging this gives:
$$ \frac{d\tau}{dt} = 1 + \frac{gx}{c^2} $$
which is just the equation (1) that we started with.
I am afraid your allegoric figuring out why gravity causes time dilation is further confusing. The physical reality is measured by the metric of a reference frame which composes the coordinates, but the single coordinates do not have necessarily a specific meaning in GR (general relativity).
The Einstein equivalence principle allows to describe gravity in terms of geometry (metric) of a curved spacetime.
If we consider a static (Schwarzschild) spherical mass, radially the curvature is different and the proper time of stationary observers, as given by the metric, is progressively slowing as you approach the mass, if compared to the time measured by an observer far away from the mass.
The time dilation of SR (special relativity) is still a different concept as it is symmetrical between two observers in uniform relative motion. Instead the gravitational time dilation is not symmetrical; in fact the stationary observer measures a time contraction if compared to the far away observer.
Best Answer
As already said in the comments, yes. However, gravitational time dilation is described by general relativity. One will find that clocks closer to a massive object (such as the earth) will tick slower compared to clocks farther away. In other word, time "flows"1 slower closer to the surface of the earth.
There are two things however that I'd like to further elaborate on.
First, you cannot tell that time has "slowed down" for you in your own frame of reference. For you, time will always appear to pass at the same "rate".
Second, I'm not sure about whether you can say that height or gravity causes time dilation. In general relativity, gravity is the curvature of spacetime (thus doesn't cause it). However, this is more a question of language – I just wanted to clarify.
1 I put this in quotes because time cannot actually flow – see What is time, does it flow, and if so what defines its direction?. Though it should be clear what is meant by that.