Why Can the Gamma Factor Exceed 2 in Relativity?

[NoahsArk]

I don't know why this problem just occurred to me- sorry if it's a silly question: Why is it that there isn't a limit to how high the gamma factor can be? How can you ever have a gamma factor which is more than 2, for example? Although if you plug the numbers into the equation for gamma it makes sense that you can have as high a gamma factor as you want, when you look at the problem visually as follows it seems impossible:

If a moving frame S¹ has a light clock aboard his ship which is one light second high, that light clock will be the same height in the stationary S frame, and the vertical side of the right triangle which represents the light clock will be one light second high. From S's point of view, the base of the right triangle can't be any longer than 1 light second because S¹'s ship can't be traveling faster than light speed. The longest the base of the right triangle can be is slightly less than 1. So, assuming that the height and base of the right triangle are equal, the hypotenuse will never be more than double the height of vertical side. Since the length that S observes the light beam traveling can never be more than double what S¹ observes, how can there ever be a gamma factor of more than 2?

 

[PeroK]

I don't know why this problem just occurred to me- sorry if it's a silly question: Why is it that there isn't a limit to how high the gamma factor can be? How can you ever have a gamma factor which is more than 2, for example? Although if you plug the numbers into the equation for gamma it makes sense that you can have as high a gamma factor as you want, when you look at the problem visually as follows it seems impossible:

If a moving frame S¹ has a light clock aboard his ship which is one light second high, that light clock will be the same height in the stationary S frame, and the vertical side of the right triangle which represents the light clock will be one light second high. From S's point of view, the base of the right triangle can't be any longer than 1 light second because S¹'s ship can't be traveling faster than light speed. The longest the base of the right triangle can be is slightly less than 1. So, assuming that the height and base of the right triangle are equal, the hypotenuse will never be more than double the height of vertical side. Since the length that S observes the light beam traveling can never be more than double what S¹ observes, how can there ever be a gamma factor of more than 2?

The obvious answer is "do the maths"!

 

[Ibix]

Light is traveling on the diagonal as viewed from S, so its vertical speed is less than c. So the up-and-down time is greater than 2s.

 

[NoahsArk]

Ok I see where my misconception is coming from. I must have had temporary brain freeze today because I understood this before. The vertical side of the right triangle is not representing the height of the light clock. It's representing the elapsed time on S¹ 's clock. Because this vertical side gets less and less the faster S¹ moves away from S, there is no limit to how many times longer the hypotenuse can become and therefore no limit to how large gamma can be.

 

[PeroK]

Ok I see where my misconception is coming from. I must have had temporary brain freeze today because I understood this before. The vertical side of the right triangle is not representing the height of the light clock. It's representing the elapsed time on S¹ 's clock. Because this vertical side gets less and less the faster S¹ moves away from S, there is no limit to how many times longer the hypotenuse can become and therefore no limit to how large gamma can be.

I would say it's the base of the triangle that is of variable length (with no upper limit). If the speed of the clock is nearly $c$, then in the "stationary" frame, the light will travel a long way horizontally before it reaches the top of the clock.