I generated so many emails in my debate on Special Relativity that I have all this content about it sitting around. I just realized that it's best that I use it as material for short blog posts that can serve as educational lessons for learning some of the basic concepts in SR. It's possible to understand SR from a conceptual framework without knowing any of the math. The math certainly helps fully understand the theory, but I think that for the general public it's at least better to know the basic ontology of what entails from SR by understanding it conceptually rather than not understanding anything at all.
For this first lesson, I will explain how time dilation is symmetric. This lesson expects you to have some basic familiarity and understanding of the concepts in Special Relativity, like an inertial frame, a light clock, a spacetime diagram, and what a worldline is, etc. It is not intended to be a full lesson from which you can learn the theory in its entirely.
In Special Relativity, time dilation is symmetric. For two inertial observers in relative motion their clocks will slow down at a rate equal to each other. Using screenshots I've taken from this video, I will explain how this works.
We first start out with a 3D representation of time in a spacetime diagram showing the position of two light clocks held by two observers moving relative to one another. One is held by Albert Einstein who is standing on a train platform, and the other is held by Hendrik Lorentz who is standing on a train moving relative to Einstein.
Image 1 above shows the relationship between the two. The zigzag pattern of the yellow lines are the worldlines of the light in their light clocks. They are the paths of the light through space and time, or spacetime.
Next we label each event where the light in the light clock reaches the mirrors at the ends of the clock and bounces back, as seen in image 2. I chose to label them 5:00, 5:01, 5:02, 5:03...etc., but this is completely arbitrary. They could be labeled with any time units and it would make no difference for the concept. The specific readings of the clocks at each spacetime point, which we call an event, will be agreed upon by all observers regardless of their relative motion. (More on this in lesson 2)
Looking at Einstein's clock from his reference frame the time units are shown as such in image 3. At 5:03, his clock and the clock held by Lorentz, who is in relative motion, pass by each other from this perspective. Suppose that at that moment both of their clocks read 5:03.
In image 4, we see that the red line indicates the length of one minute for Einstein from his reference frame, whereas the blue line indicates the length of one minute for Lorentz. The blue line is longer because from Einstein's reference frame, Lorentz's clock is in relative motion and its relative motion dilates its reading of time. Lorentz's clock therefore moves slower than Einstein's clock from Einstein's reference frame. This is due to the constancy of the speed of light, one of the two postulates in Special Relativity. Since Lorentz's light clock is moving relative to Einstein, the light beam has to travel longer distances as it bounces up and down between each mirror in the light clock, and that results in its ticks being slower. This is the meaning of time dilation. It is a direct manifestation of the constancy of the speed of light.
But what would Lorentz observe from his reference frame? Will he see Einstein's clock as moving faster since his clock is moving slower? Or will he see Einstein's clock as moving slower?
If we change perspectives to see the scenario how Lorentz would see it, we find that Lorentz would see that Einstein's clock is moving slower, not faster. In image 5, Lorentz is now still and Einstein is now in relative motion and moving away from him. This is because in Special Relativity there is no such thing as absolute motion or rest. Motion and rest are always relative to other things. In this scenario, Lorentz's motion was relative to Einstein, who is still relative to Lorentz in image 4, and in image 5, Einstein's motion is relative to Lorentz, who is still relative to Einstein. The blue line indicates the length of one minute for Lorentz, and the red line indicates the length of one minute for Einstein. Notice that Einstein's red line is now longer than Lortentz's blue line. And notice that it's longer by exactly the same length as Lorentz's was to Einstein's in image 4. This shows how time dilation is symmetric to relative observers in inertial (non-accelerating) frames. When you change perspectives from each relative observer's reference frame, the amount of time dilation each observe of the other is exactly the same.
To appreciate this in a fuller perspective, consider what it would look like from Einstein's perspective again in image 4, but with the t and x axis of a spacetime diagram with Lortenz's reference frame drawn.
In image 6, the simultaneity planes of each observer when their clock reads 5:04 are drawn, in red for Einstein, and blue for Lorentz. Events along the red line are considered simultaneous by Einstein, and events along the blue line are considered simultaneous by Lorentz. If you notice, for Einstein at 5:04, his red line that indicates what he considers "now" cuts into Lorentz's worldline before it has hit 5:04. It is only 5:03:50 at that time. And notice that for Lorentz at 5:04, his blue line that indicates what he considers "now" cuts into Einstein's worldline before it has hit 5:04 by exactly the same amount as Einstein's did to him. It is only 5:03:50 at that time on Einstein's clock. This explains why time dilation is fully symmetric. If you and I are in relative non-accelerating motion between each other, if my clock is time dilated by 20% according to you, then your clock will be time dilated by 20% to me.