As I was interested I decided to derive the time dilation equation. My video if you're interested in doing so yourself is linked here.

To begin with I took 2 observers, one stationary and one moving. Both use a photon clock to measure the passage of time. It's just two parallel plates, when a photon is emitted and then reflected and received a tick of the clock is recorded.ย

If you imagine the moving observer moving past the stationary observer at a velocity "v" imagine what happens according to the stationary observer in terms of the moving photon clock. If it isn't apparent to you straight away the photon moves in a triangular path:

ย The values annotated here are simple from the fact that displacement in a given time is equal to the speed multiplied by that time. Here by "t" I mean the time it appears for a tick of the moving clock according to the stationary observer.

From there you can use Pythagoras's theorem to reach the following conclusion:

Manipulating this a bit you get:

Multiply by 4:

Move the "v squared t squared" to the other side and factor out t:

Divide by the brackets on the right:

Square root both sides:

Factor out c squared from the square root (this step seems useless but really isn't)

The 2d/c ends up being a tick of the stationary clock. So we can call this "t zero" and substitute in to relate a tick of the stationary clock to a tick of the moving clock:

**And that's it!!**

Time dilation equation derived easily. I encourage you to watch my video on this subject explaining each step.

Thanks to Saeed a PlanckTime user and friend for suggesting this post. He also did this derivation and I encourage him to post his work here too!

PlanckTime - Amin

Hey, my working had some mistake (its nothing major, and its just that i wrote something down wrong) but it is otherwise the same. Here is my beloved board with the proof attatched. I'll also attatch the next little bit which is length contraction from special relativity. And i suspect that a video will be made on it soon (coughvideosuggestioncough)

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"Either this wallpaper goes, or I go"- Oscar Wilde's last words

The beloved board strikes again XD. Thanks Saeed!

PlanckTime - Amin

Just wait until i get the portable whiteboard out, then stuff gets REALLY interesting ๐

"Either this wallpaper goes, or I go"- Oscar Wilde's last words

Can't wait !!!ย

PlanckTime - Amin

Out of curiousity, will you make a video about the length contraction?

"Either this wallpaper goes, or I go"- Oscar Wilde's last words

100% will do Saeed. Time is precious but look out for it in the near future!

PlanckTime - Amin

Maybe i should go move at reletavistic speeds so that i can get it earlier ๐

ย

"Either this wallpaper goes, or I go"- Oscar Wilde's last words

Maybe i should go move at reletavistic speeds so that i can get it earlier ๐

The banter is too strong ahahah XD

PlanckTime - Amin