I’ve been going down a physics rabbit hole with the speed of light on this sub and am honestly blown away. I understand now that the faster you move the slower you experience time.
Is this relationship directly proportional?
i.e. moving at half the speed of light makes you observe time at half the normal speed.
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First of all its important to note that all of this is relative, and in your own frame you always experience time as moving at the same rate.
Time dilation is the change in the interval between events between you and another reference frame moving relative to you, e.g half the speed of light.
To answer the question, it’s not linear. The equation for time dilation is t’ = γt, where γ (gamma) is the Lorentz factor
γ = 1/sqrt(1-v^2/c^2).
This doesn’t vary linearly, for small values of v this is effectively just equal to 1 and there is no time dilation, as the velocity approaches the speed of light this equation approaches infinity. [This graph](https://upload.wikimedia.org/wikipedia/commons/thumb/0/00/Lorentz_factor.svg/2019px-Lorentz_factor.svg.png) shows how gamma changes as you approach the speed of light. For v=0.5c, gamma is about 1.15 so there would be about 15% time dilation.
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