- #1
Arkalius
- 72
- 20
Hello everyone. I've just recently found this forum and it has been a lot of fun browsing around. I've recently taken a stronger interest in the topics of relativity in physics and have recently developed a much better understanding of SR (and somewhat of GR) than I'd had previously and its been fun exploring that understanding in various scenarios.
Of greater interest to me recently is how things appear to observers in relativistic situations when you consider the travel time of light. Many thought experiments and scenarios are described from the "measured" viewpoint, or as if each observer could witness all instantaneous events (from their frame) at the same moment. This has its uses, and it simplifies an already complicated subject, but I find it useful to explore what observers would actually see in reality.
To that end I'd come up with some simple equations that give a ratio of observed length and speed to the actual length and speed of an object moving toward or away from you. These are $$\begin{align} \frac 1 {1-\beta} \\ \frac 1 {1+\beta} \end{align}$$ with (1) being for objects moving toward you, and (2) for objects moving away, and with ##\beta = \frac v c##. These were great and all, but I wanted a more general equation that worked in 3d and 4d spacetime. In those, an object doesn't always move directly toward or away from you.
So, I came up with this more generic equation for this ratio: $$\gamma^2 \left( \beta \cos \alpha + \sqrt{1-\beta^2 \sin^2 \alpha} \right)$$ Here, ##\alpha## is the (actual) angle between the relative velocity vector and position vector. You will see that when that angle is 0 or ##\pi##, the equation reduces to the two I have above. You simply multiply actual length or velocity by this value to get the observed length or velocity.
Another useful equation is for the observed angle of deflection and that is given by $$\alpha_{obs} = \alpha - \arcsin \left( \beta \sin \alpha \right)$$
I'd not seen equations like these listed anywhere that I'd looked in the past, and I found them useful for examining scenarios for how they would appear to the observers. It certainly brings a different perspective to the situation. Things moving toward us won't appear contracted, but rather stretched out. Also, things can appear to move toward us at many times the speed of light because of this. It also means nothing can appear to move directly away from us at more than half the speed of light either.
Anyway, I look forward to participating in more discussions on this forum and learning more interesting things about relativity and other topics.
Of greater interest to me recently is how things appear to observers in relativistic situations when you consider the travel time of light. Many thought experiments and scenarios are described from the "measured" viewpoint, or as if each observer could witness all instantaneous events (from their frame) at the same moment. This has its uses, and it simplifies an already complicated subject, but I find it useful to explore what observers would actually see in reality.
To that end I'd come up with some simple equations that give a ratio of observed length and speed to the actual length and speed of an object moving toward or away from you. These are $$\begin{align} \frac 1 {1-\beta} \\ \frac 1 {1+\beta} \end{align}$$ with (1) being for objects moving toward you, and (2) for objects moving away, and with ##\beta = \frac v c##. These were great and all, but I wanted a more general equation that worked in 3d and 4d spacetime. In those, an object doesn't always move directly toward or away from you.
So, I came up with this more generic equation for this ratio: $$\gamma^2 \left( \beta \cos \alpha + \sqrt{1-\beta^2 \sin^2 \alpha} \right)$$ Here, ##\alpha## is the (actual) angle between the relative velocity vector and position vector. You will see that when that angle is 0 or ##\pi##, the equation reduces to the two I have above. You simply multiply actual length or velocity by this value to get the observed length or velocity.
Another useful equation is for the observed angle of deflection and that is given by $$\alpha_{obs} = \alpha - \arcsin \left( \beta \sin \alpha \right)$$
I'd not seen equations like these listed anywhere that I'd looked in the past, and I found them useful for examining scenarios for how they would appear to the observers. It certainly brings a different perspective to the situation. Things moving toward us won't appear contracted, but rather stretched out. Also, things can appear to move toward us at many times the speed of light because of this. It also means nothing can appear to move directly away from us at more than half the speed of light either.
Anyway, I look forward to participating in more discussions on this forum and learning more interesting things about relativity and other topics.