Because it is a right triangle, it is very simple to move along the direction of AC or BC (only the relative speed is mentioned in the question, not the direction of movement), because only one high frequency is changed. In fact, it can be proved that when the dimension is compressed in any direction, the area is compressed in the same proportion.
√( 1-v？ /c？ )=0.6,γ=5/3
So the area seen by the observer is 5/3 times of the original area.
We need to pay attention to a very important question, which is not clearly stated in the textbook. The conclusions in the theory of relativity are all relative to the observation results of observers, but they are often expressed in an ambiguous way and sometimes in the opposite way.
For example:
In the expression of time, the expression of relativity is that "the time on the high-speed moving system is slower than that of the observer", which is the direct expression of the observer's observation results. The actual meaning is "the true value is less than the observed value".
In terms of mass expression, the expression of relativity is that "the mass of a high-speed moving object becomes larger relative to the observer", which is an antonym expression relative to the mass of the object. The actual meaning is that "the true value is less than the observed value".
The same time expression, it should be said that "the mass of high-speed moving objects becomes smaller."
This is very important, if you don't pay attention, it will cause confusion. It is best to emphasize whether it is the value seen by the observer or the real value converted from the observed value.
For a triangle moving at a high speed relative to the observer, according to the principle of relativity, the true value becomes smaller when the relative observation value is unchanged. As far as absolute space-time is concerned, the observed value is greater than the true value.
These words are said because the length of each side of the triangle given in the title is only marked with AB, AC and BC, and it is not stated whether the so-called area refers to the static area of the triangle or the area seen by the observer. Fortunately, relativity doesn't care about this, and the calculation result is only a proportion. Namely: observed value/true value = 5/3.
In addition to say a few words:
This is because it is found that the formula of special relativity used in different textbooks is different. Some books only give the formula of absolute value, without adding the observed value caused by the optical path difference problem, because this score is not caused by the relativistic effect. The formula given in some books is a formula with optical path difference, that is to say, it is not an absolute value, but a real observation result caused by the time difference caused by the propagation speed of light. This result will produce different values because of the different direction of motion.
Personally, I think the relativity factor in the theory of relativity (Lorenz doesn't like others to call that factor "Lorenz factor"). There is no factor to calculate the optical path difference, so it should be calculated as a parameter without optical path difference. The optical path difference does not belong to the relativistic effect, and should be superimposed in the result.
The above picture shows Lorenz's model for deducing Lorenz factor. In the figure, A is a system moving at a speed v relative to O, and B is a point on the A system. While A and O overlap one another, a photon is emitted from A to B. In A, the path of light is ct', in O, the path of photon is CT, and A is shifted by vt in T time.
It does not consider the time required for light to travel from the current position of A to O, so it is an absolute value without optical path difference.
(The time for a photon to travel from B to A and from B to O is twice that of a one-way trip, so there is no time difference in terms of time ratio. )
According to Pythagorean Theorem (ct')? +(vt)？ =(ct)？ ;
Solution: t' = t √ (1-V? /c？ ),
According to the relative speed equivalence: that is, the speed of A relative to O is V, and the speed of O relative to A must be V, we can know that the distance of vt is vt' in A's view. So the ratio of distance to time is the same, s' = s √ (1-v? /c？ )。
As can be seen from the figure, no matter what V is, it is ct that is affected, and ct' will not change because of the change of V. We call the constant parameter "true value" and the value that will change because of some factors (such as speed) is called "observed value" (such as T).