We previously derived the Lorentz transformation and the velocity addition formula through matrix transformations. It must be said that the matrix derivation method has a fascinating charm. Today, in books covering relativity (including electrodynamics and general relativity), the tool of tensors has mathematically replaced it. This is actually a generalization of matrices (as mentioned before, a second-order tensor is equivalent to a matrix). Adopting such a form is because it fully reflects the symmetry and transformation relationships of relativity. This article will discuss some basic conclusions of special relativity, including the relativity of simultaneity, length contraction, and time dilation.
In this article, the speed of light is c=1.
Relativity of Simultaneity
In the same spacetime, by taking two spacetime coordinates for a Lorentz transformation and then taking their difference, we obtain: \left[\begin{array}{c} \Delta x\\ \Delta t \end{array}\right]=\frac{1}{\sqrt{1-v^2}}\left[\begin{array}{c c}1 & v\\ v & 1 \end{array}\right]\left[\begin{array}{c}\Delta x'\\ \Delta t' \end{array}\right]
What does the above equation tell us? We find that if \Delta t'=0, it does not imply \Delta t=0. This indicates that two events occurring simultaneously in one reference frame are not simultaneous as seen from another inertial reference frame due to the difference in location (\Delta x' \neq 0). This is the relativity of simultaneity. It tells us that in relativity, if we want to talk about events happening simultaneously, we must specify in which reference frame.
Length Contraction (Meter-Stick Contraction Effect)
Since even simultaneity is relative, one can naturally guess that in relativity, nothing is absolute (of course, some invariants still exist, the most obvious being the invariance of the speed of light). In fact, the length and time we usually see are not absolute; they vary depending on the reference frame. When we usually talk about length and time, we intuitively think they are just intrinsic properties of objects and are invariant, so we do not think about measurement issues. However, in relativity, due to relativity, measurement must be defined before its changes can be described quantitatively.
First, let’s look at length. How do we measure length? We cannot always place a ruler on the object being measured, but we can do this: establish a coordinate system belonging to our reference frame and then measure the coordinates of the object. The length is obtained by subtracting the coordinates. That is:
Measuring the length of an object in the S reference frame:
At the same moment, record the coordinates x_1 and x_2 of the head and tail ends of the object, and find |\Delta x|.
Thus we have: \left[\begin{array}{c} \Delta x\\ 0 \end{array}\right]=\frac{1}{\sqrt{1-v^2}}\left[\begin{array}{c c}1 & v\\ v & 1 \end{array}\right]\left[\begin{array}{c}\Delta x'\\ \Delta t' \end{array}\right]
Solving the system yields: \Delta x=\sqrt{1-v^2} \Delta x'
Where \Delta x' is measured in the S’ reference frame which is stationary relative to the measured object—in other words, the length of the object when it is at rest; while \Delta x is measured in the S reference frame which is moving relative to the object—that is, the length of the object when it is in motion. Therefore, it can be summarized in one sentence:
The length of an object contracts in the direction of motion!
For example, if a spacecraft has a length of 100m when stationary, as long as it flies fast enough, we might see (measure) a spacecraft with a length of only 1m.
Time Dilation (Clock Slowing Effect)
Time must also be considered using a similar method. Of course, we must first define how to measure time:
Measuring the time interval of the same event in the S’ system from the S reference frame:
Record the time coordinates t_1 and t_2 before and after the same event occurring at the same location in the S’ system, and find \Delta t.
It must be emphasized that it is an event at the same location in S’, such as the event of an astronaut in a spacecraft waving to us. This actually requires \Delta x'=0, then: \left[\begin{array}{c} \Delta x\\ \Delta t \end{array}\right]=\frac{1}{\sqrt{1-v^2}}\left[\begin{array}{c c}1 & v\\ v & 1 \end{array}\right]\left[\begin{array}{c} 0\\ \Delta t' \end{array}\right]
It quickly follows that: \Delta t=\frac{1}{\sqrt{1-v^2}} \Delta t'
That is to say:
Time in motion slows down!
As long as the spacecraft moves away from us at a sufficiently high speed, an astronaut on the spacecraft waving to us might only take 2 seconds in their frame, but it might take 1 hour in our view!
Summary
By now, our view of spacetime in special relativity has basically been established. From a formalist perspective, we should also add a geometric description of relativistic spacetime, but we do not emphasize that here. From a physical perspective, we still lack the content of relativistic dynamics, which may be discussed slowly in future articles. This article is not a relativity textbook, but rather some of my more intuitive views while learning relativity, so for more fundamental content, please read relevant works. In the next article, we will see that relativistic effects are not necessarily only manifested in high-speed particles; the beauty of relativity is also reflected in very slow motion.
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