# Time and gravity

am i right in saying that if you could raise the distance in the speed = distance/ time equation without altering the other parameters it would give the appearance that (to an outside observer) time would appear (on paper) to slow? is this how we come to the conclusion that time near a massive body runs slower because space is distorted by gravity altering the parameters of this equation?

Please note that your speed=distance/time equation is just that, an equation. Being an equation, I could then write $vt=d$ (I've only moved time to the other side). Notice that if you stipulate that speed and time parameters are kept constant, that means the distance parameter MUST be constant. The left-hand side of the equation is equal to the right-hand side. If one side is constant, so is the other. Long story short, this is not how we concluded that gravity dilates time.

Gravitational time dilation has many forms, but the best one comes from something called the Schwarzschild metric. A metric is an equation that represents how each dimension (length, width, depth, duration) relates to each other dimension. The Schwarzschild metric (named after the guy) is just a metric with an object of mass $M$ located at the center of your coordinate system.

I'll spare you the boring details (well, I think they're fun, but I'm pretty sure that's because I'm crazy) but we can directly pull a time term from this metric that shows the following:

$$d\tau^2=\left(1-\frac{2GM}{c^2R}\right)dt^2$$

$d\tau$ is an amount of time that passes for an observer a fixed distance $R$ away from an object of mass $M$. $dt$ is the relative amount of time for an observer very far away from the mass. $G$ is Newton's gravity constant and $c$ is the speed of light. I'll spare you the messy derivation of this. Suffice it to say that this is where we get gravitational time dilation from; not some notion of gravity distorting only one parameter of $v=d/t$