> Is travelling at the speed of light, actually travelling a a fraction of the speed of time?
This is probably one of the more counter-intuitive simple calculations you can do in physics.
In special relativity, distance is given by:
ds^2 = dx^2 + dy^2 + dz^2 - (c^2)dt^2
if X is your total distance in space, you have:
dX^2 = dx^2 + dy^2 + dz^2
Which is just the standard Pythagorean theorem of Euclidean geometry.
Further, your velocity is given by dX/dt. If you are traveling at the speed of light, you have:
dX/dt=c
From which you can derive
dX^2 = (c^2) dt^2
dX^2 - (c^2) dt^2 = 0
ds^2 = 0
In other words, the "distance" light travels in space time is 0.
> just curious... does "time" have a "speed" ?
It is not clear how to parse this question. Traditional "speed" is defined as distance over time. We can give this meaning for time itself by realizing that there is no single notion of time in relativity. As such, you could consider the line parallel to the time axis in the coordinate system of observer A. Since dt=0 in the coordinates of observer A, the speed of this line is not well defined. However, we could consider the coordinates of observer B. Assuming B is moving relative to A, he would see this line as being slanted, with both a time component, and a space component. As such, B could compute the speed of this line as dX'/dt', where X' is the total displacement along B's 3 spatial dimensions, and dt' is the displacement in B's time dimension. As such, B could meaningfully answer "what is the speed of A's time". Assuming I didn't mess up on the math, dX'/dt' turns out to be the velocity of A relative to B. This is a curious result that I have never seen before, but I can't really see any physical significance to it.
B could also compute dt/dt', where t' is the time axis in B's coordinate system. This computation seems more useful as it gives a direct measure of time dilation. Unsurprisingly, it also works out to be the Lorentz factor.
This is probably one of the more counter-intuitive simple calculations you can do in physics.
In special relativity, distance is given by:
if X is your total distance in space, you have: Which is just the standard Pythagorean theorem of Euclidean geometry.Further, your velocity is given by dX/dt. If you are traveling at the speed of light, you have:
From which you can derive In other words, the "distance" light travels in space time is 0.> just curious... does "time" have a "speed" ?
It is not clear how to parse this question. Traditional "speed" is defined as distance over time. We can give this meaning for time itself by realizing that there is no single notion of time in relativity. As such, you could consider the line parallel to the time axis in the coordinate system of observer A. Since dt=0 in the coordinates of observer A, the speed of this line is not well defined. However, we could consider the coordinates of observer B. Assuming B is moving relative to A, he would see this line as being slanted, with both a time component, and a space component. As such, B could compute the speed of this line as dX'/dt', where X' is the total displacement along B's 3 spatial dimensions, and dt' is the displacement in B's time dimension. As such, B could meaningfully answer "what is the speed of A's time". Assuming I didn't mess up on the math, dX'/dt' turns out to be the velocity of A relative to B. This is a curious result that I have never seen before, but I can't really see any physical significance to it.
B could also compute dt/dt', where t' is the time axis in B's coordinate system. This computation seems more useful as it gives a direct measure of time dilation. Unsurprisingly, it also works out to be the Lorentz factor.