Simple Derivation of the Lorentz Factor (γ)

The Lorentz Factor is the factor by which time is dilated or length is contracted due to relativistic motion. It is commonly represented by the Greek letter Gamma (γ), and there is a surprisingly simple derivation for it. Of course, this derivation assumes some prior knowledge in basic special relativity, but if you didn't have that, you wouldn't be watching this video in the first place.

[kaitlinpmccreery]

It's amazing how hard it is to find an explanation and walk-through of this derivation. Thank you so much for this contribution!

[gregchristos]

Literally hundreds of videos posted on this subject and this is the first that neatly and perfectly explains the derivation

[PurpleChevron]

Phenomenal. Straight, simple and to the point. Bravo.

[christianjulianto]

aaahh finally some one can tell where exactly gamma come from . i've spent hours of time to learn theory of relativity and everything so clear in another website but no one mention where the fck is gamma come from . thanks now i can continue my study happily .

[sayedelghairb]

in my opinion, this is the best derivation. concise and simple at the same time. Good luck!

[thehighground]

um THANK YOU for making this so much clearer than all of my textbooks! This was such a wonderful explanation, for visual learners especially!!

[thomasconrod]

This is the clearest description of the algebra involved. However some people trying to get this for the first time need a bit more illustration on the movement of the light clock from observer's reference frame. Thank for your clear and complete description of the relatively simple math. It's cool to see a rather esoteric concept explained pretty simply by high school geometry.

[ouiner]

Dude thank you so much this is the clearest and easiest explanation ive been able to find

[ATorres]

That was more suspenseful and satisfying than watching a David Blaine performance, thank you for taking the time to review gamma so succinctly!

[sunjayroy]

You've assumed that time dilates before deriving the gamma factor, which is what the gamma factor is supposed to prove. Surely that's an error in logic?
Also just because the source is moving doesn't mean the emitted light would be observed at an angle due to the vector addition of velocities.

[cardayz]

Under what assumptions can we sue this derivation? It feels so simple and I was wondering if there are assumptions we must make to have this derivation make sense

[cyan]

Holy shit that's actually great. When in doubt. Check the math.

[shaili]

Amazing explanation, finally a satisfying end to the twin paradox.

[GAR]

surely the photon only appears to travel a longer distance? It was emitted from the starting point and continued to travel in the same direction?

[atlisveinbjornsson]

Point S is moving and it emitts a photon, observer isn't moving so it sees S moving but the photon just travels in straight direction from point S, so shouldn't observer just see the photon move straight up?

[GaudioWind]

Ok, but I'd like to ask if the draw is not a little inaccurate. The way it is, it looks like v is even greater than c, isn't it? I suppose that ct should be represented greater than vt', right?

[Suesco]

this still doesnt make sense when referring to the idea that the two observers clocks go out of sync. If you flip the visual around, and have O send the photon, and S be observing it, then you get the dilation in reverse. Also it doesnt take into account the blue shift/length contraction caused by the lateral momentum. the only interpretation that could be correct is each observer splitting into a new universe with every change of state

[severyn]

Why did you distribute power to each individual letter? Should it be that ct' ct and vt' are the names of the sides of this triangles only that in fact are the lengths measurements. Thus if you separate them and square them individually it will be like: c×c×t×t which is doubling the amount of ct. Because (ct) × (ct) is not the same as c×c×t×t due to the fact that c and t are different units of measurement, one is speed and one is time. If you had only time there then it would be fine. If you had only speed there it would be fine too. But you have speed × time= distance. Also Pythagoras theorem is C^2=A^2+B^2. It is a single letter equation that represent the lengths of sides of that triangle. When you use two letters separately c and t or v and t this means you have not one triangle but two triangles that are joined together. You can do the same extraction for c or v just like for t or t'. However what I noticed is that the whole transformation is slightly useless as you don't have to transform the Pythagoras theorem at all if you know the lengths of two sides ct and vt'. Simply do the math using Pythagoras theorem for ct' and then when you know how long that third longest side is, do the math for time needed for light travel that distance: |ct'|:c=T. Also the result of that equation for time dilation represents not dilation of time but the lenght of that longest side of that triangle measured in time units. Obviously if this is the longest side of triangle then t' will always be different than t. You can't have any other result from this geometrical equation.

[markdallen]

Why is the ct on the left of the graph not ct' ?

[abedbbb]

the math is fine but the graphic representation does not fit, if we do an anology of photon to a projectile, a projectle would never take a path like described here why the photon had to have a inclined path instead of direct path once it is emitted from the first position ?