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This is something I've been meaning to write for a while, I finally got the time to do it, though the topic isn't currently "hot". I'm sure it will pop up again, though.
Imagine we have two observers, moving at a constant velocity relative to each other far away from any objects that might perturb their motion. We will call the two observers Adam (abbreviated as A) and Beth (abbreviated as B). Each of these observers carries an idealized clock - for the purposes of this thought experiment, a physical atomic clock is close enough to ideal as makes no difference.
One of the results of special relativity is that in Adam's frame of reference, Beth's clock runs slow - and vica versa. Formally, we can say the following. Along A's trajectory, which is also called A's worldline, every unique reading on A's clock corresponds to a unique event. The same applies to B's worldine, every clock reading defines a unique event. The concept of simultaneity means that we can define a 1:1 mapping between the events on A's worldine and the events on B's worldline that happen "at the same time" in A's frame of reference.
Example: Suppose we have a time dilation factor of 2:1. Then in A's frame of reference, the event on A's worldline where his clock reads 1 is simultaneous with the event on B's worldline where her clock reads 1/2.
We can write a general formula for this, for ease of exposition we will continue to assume the time dilation factor is 2:1 as in our example. The argument can easily be generalize to other time dilation factors, however. When A's time reads X, B's time reads X/2, where X is any number. Equivalently we can also say that when B's clock reads X, A's clock reads 2X using basic algebra.
Now, the concept of symmetrical time dilation implies that in B's frame of reference, A's clock is the clock that runs slow. So in B's frame of reference, when B's clock reads X, A's clock reads X/2 Or, equivalently, when A's clock reads X, B's clock reads 2X
Now, let's try to reconcile these facts. In A's frame of reference, when A's clock reads X, B's clock reads X/2. In B's frame of reference, when A's clock reads X, B's clock reads 2X. We note that 2X is not equal to X/2. So, what event on B's woldline is "really" simultaneous with the event on A's worldline where A's clock reads 1?
The answer is actually very simple. Simultaneity in relativity is relative, so both statements are true. One is true in A's frame of reference, the other is true in B's frame. Thus, "at the same time" in A's frame of reference is not the same concept (or mapping) as "at the same time" in B's frame of reference. In Newtonian physics, time is absolute, so the concept of "at the same time" is universal and not dependent on the observer. This is not the case with relativity, and symmetrical time dilation is just one of the consequences.
The concept of the relativity of simultaneity was popularized by Einstein in "Relativity: the special and general theory" in chapter 9, in a thought experiment involving a train. This is currently available online (and has been for some time) at http://www.bartleby.com/173/9.html.
Another paper on the topic that I find useful (I've had little reader feedback) is
"The challenge of changing deeply held student beliefs about the relativity of simultaneity", by Scherr, Shaffer, and Vokos. The abstract of this paper is on arxiv at http://arxiv.org/abs/physics/0207081. The full pdf text is usually available at http://arxiv.org/ftp/physics/papers/0207/0207081.pdf.
Imagine we have two observers, moving at a constant velocity relative to each other far away from any objects that might perturb their motion. We will call the two observers Adam (abbreviated as A) and Beth (abbreviated as B). Each of these observers carries an idealized clock - for the purposes of this thought experiment, a physical atomic clock is close enough to ideal as makes no difference.
One of the results of special relativity is that in Adam's frame of reference, Beth's clock runs slow - and vica versa. Formally, we can say the following. Along A's trajectory, which is also called A's worldline, every unique reading on A's clock corresponds to a unique event. The same applies to B's worldine, every clock reading defines a unique event. The concept of simultaneity means that we can define a 1:1 mapping between the events on A's worldine and the events on B's worldline that happen "at the same time" in A's frame of reference.
Example: Suppose we have a time dilation factor of 2:1. Then in A's frame of reference, the event on A's worldline where his clock reads 1 is simultaneous with the event on B's worldline where her clock reads 1/2.
We can write a general formula for this, for ease of exposition we will continue to assume the time dilation factor is 2:1 as in our example. The argument can easily be generalize to other time dilation factors, however. When A's time reads X, B's time reads X/2, where X is any number. Equivalently we can also say that when B's clock reads X, A's clock reads 2X using basic algebra.
Now, the concept of symmetrical time dilation implies that in B's frame of reference, A's clock is the clock that runs slow. So in B's frame of reference, when B's clock reads X, A's clock reads X/2 Or, equivalently, when A's clock reads X, B's clock reads 2X
Now, let's try to reconcile these facts. In A's frame of reference, when A's clock reads X, B's clock reads X/2. In B's frame of reference, when A's clock reads X, B's clock reads 2X. We note that 2X is not equal to X/2. So, what event on B's woldline is "really" simultaneous with the event on A's worldline where A's clock reads 1?
The answer is actually very simple. Simultaneity in relativity is relative, so both statements are true. One is true in A's frame of reference, the other is true in B's frame. Thus, "at the same time" in A's frame of reference is not the same concept (or mapping) as "at the same time" in B's frame of reference. In Newtonian physics, time is absolute, so the concept of "at the same time" is universal and not dependent on the observer. This is not the case with relativity, and symmetrical time dilation is just one of the consequences.
The concept of the relativity of simultaneity was popularized by Einstein in "Relativity: the special and general theory" in chapter 9, in a thought experiment involving a train. This is currently available online (and has been for some time) at http://www.bartleby.com/173/9.html.
Another paper on the topic that I find useful (I've had little reader feedback) is
"The challenge of changing deeply held student beliefs about the relativity of simultaneity", by Scherr, Shaffer, and Vokos. The abstract of this paper is on arxiv at http://arxiv.org/abs/physics/0207081. The full pdf text is usually available at http://arxiv.org/ftp/physics/papers/0207/0207081.pdf.
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