# Length contraction and time dilation relationship

We did the following derivation in my electromagnetism lecture

Observer B measures $$c\Delta t'=d$$

Observer A measures $$c\Delta t =l$$

From Pythagoras' theorem: $$d^2+(v\Delta t)^2=l^2$$ $$(c\Delta t')^2+(v\Delta t)^2=(c\Delta t )^2$$ from which $$\Delta t =\Delta t'\gamma\qquad(\alpha)$$
with $$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

Then for the bar moving to the right with the same velocity of the B reference frame, B measures its length as $$L'$$ and A measures it as $$L$$ $$\Delta t' =\Delta t\gamma \qquad(\beta)$$ $$L'=c\Delta t'=c \gamma \Delta t=L \gamma$$ that is, we get length contraction

Equation$$(\beta)$$ is what troubles me. Why don't they use equation $$(\alpha)$$ instead?

Consider the definition of length. We may define it as the difference of the spatial coordinates between two points with respect to a reference frame, when it's temporal coordinates are the same. Basically length is the distance between two points when they are at the same time coordinate.

Thus the relationship between length with respect to two different reference frame may be obtained using the Lorentz Transformation for the spacial coordinates.

Consider, the following scenario, two reference frames A and B. A is stationary, while B is moving with a constant velocity v with respect to A. We assign x and t to the reference frame A, x' and t' to reference frame B.

Note I will be taking $$c=1$$ to the end.

Now consider at some t', we measure a length l with respect to B. The length will be equal to

$$l=(x'_p-x'_o)$$

Now let's write down the transformation equations for $$x'_p$$ and $$x'_o$$.

$$x'_p=\frac{x_p-vt}{\sqrt{1-v^2}}$$

And

$$x'_o=\frac{x_o-vt}{\sqrt{1-v^2}}$$

Now we know how x transforms and hence we will substitute them into our transformation equations.

Here I will consider a simple example, however it can be generalised further. For simplicity, let's consider measurements made at t'=0, between the points x'=0, and x'.

$$x'=\frac{x-vt}{\sqrt{1-v^2}}$$

Now as we have taken $$t'=0$$, we may imply that $$t=vx$$ and from here we can rewrite our transformation equation as

$$x'=\frac{x-v^2x}{\sqrt{1-v^2}}$$

This is going to give us

$$x'=x\sqrt{1-v^2}$$

Adding the speed of light in we get

$$x'=x\sqrt{1-v^2/c^2}$$

Similarly you can arrive at the formula for tube Dilation

Now by our definition what is x'? Length as measured by the moving frame, what is x? Length as measured by the stationary frame. The key here is in the definition of length and time, which brings in the concept of length contraction, and time dilation

Also you must remember that in Special Relativity we don't use time dilation and length contraction, instead we focus on Lorentz Transformations, which are the fundamental equations of Special Relativity.

Also here is a special relativity space time graph, which works on the principles of Lorentz transformations, which will physically show how the effects take place

I know light doesn't travel at a speed relative to it's source across deep space, because high speed events like supernovae are seen in their real time span of a few days instead of being spread out over many years. If the light was travelling just 1% faster from the material blasted towards us than from material blasted the other way, and the event occurred just 10,000 light years away, it would appear spread out over 100 years by the time the light got here. If it was 10 million light years away in a neighboring galaxy, it would appear spread out over 100,000 years. Also, the time delay to correlate signals from different dishes in a very large array radio telescope is the same for a red shifted (rapidly receding) object, as a non red shifted one at the same angle. BUT, light does appear to act as if it were travelling relative to it's source (at least in our neighborhood). Using that assumption, and that the apparent change in speed of time is an optical illusion, every formula fits exactly: Lorentz Contraction (Pythagoras's Theorem), time dilation, Doppler Effect, not only for blue & red shift, but everything the object is doing including moving will appear sped up or slowed down due to the Doppler Effect compressing or spreading out whole events like it does, waves (emission of a wave is an event), and compression & elongation because the near end is seen moving first, as the light has less distance to go and arrives first. See galaxy example here: (In the ladder and barn paradox what happens if we leave the doors closed?), and the right side animation on Tyrell Rotation here:www.youtube.com/watch?v=BTY74hYdG4s. The lines coming towards you appear longer than the stationary ones. At 90° or 270°, like a rock thrown at a pole from a moving vehicle as it passes, the light coming out in a backward direction travels toward the observer, making it look turned away, and also, slowed down, red shifted and shortened (because the light from the back arrives 1st) exactly according to the Lorentz formula. In 2006, I derived a formula (in Diagram 3), that predicts how an observer should see an object, from any angle, whether the object is moving at high speed, or the observer is moving, and it makes no difference, exactly according to the Theory of Relativity. As you can see in Diag. 4, the formula actually becomes the Doppler formula at 0° & 180°, and the Lorentz Contraction formula at 90° & 270°. It may not be what is actually happening, but it is certainly food for thought, even for the most devout Einsteinians. I challenge the Einsteinian high priests to prove my maths wrong before before deleting it. 😉

• The speed of light in vacuum is always c, regardless of the speed of the source or the observer and regardless of whether the source is "in deep space" or "in our neighborhood". This is an empirical observation well supported by experiment. Commented Dec 1, 2022 at 1:29
• @Eric Smith Maybe so. The light from all parts of a supernova certainly has to travel at the same speed to avoid it appearing spread out over many years, but some aspects of relativity don't seem to make sense, like, if I'm travelling head first at relativistic speed, I get shorter. If I stand up, I get thinner. If I actually got shorter going head first, the effect of the light from my head reaching an observer in front before the light from my feet, would have more effect than the Lorentz contraction, and make me look longer. I believe Lorentz contraction is only seen from the side. Commented Dec 2, 2022 at 0:11
• @ Eric Smith I'm not saying that what is in my post is definitely the case. I am just giving people something to think about. The science bible could have some misinterpretations in it. I'm keeping an open mind. I am an atheist when it comes to spooks, and an agnostic here. I don't accept things as gospel unless they make sense, or there is compelling evidence. Commented Dec 2, 2022 at 0:29
• Just to be clear: if you're travelling head first you only "get shorter" as measured by a "stationary" observer; you yourself will not notice any difference. The same effect happens if you place two rulers at angles to one another: each measures the other to be "shorter". Similarly length contraction is a geometric effect of moving through spacetime. Commented Dec 2, 2022 at 1:02
• There's an important difference between what an observer literally "sees" (in the sense of what light hits their eyes) and what they "measure" or "observe"; the latter terms generally refer to what the final results are calculated to be after taking into account the propagation speed of the light and relative positions and speeds of objects. In discussions of relativity it is the second sense of "observe" which is almost always meant. In your scenario what they would literally see could be quite bizarre (google Terrell Rotation for details) but they would calculate you to be contracted. Commented Dec 2, 2022 at 2:08

This is the problem with theory of relativity, from definition of time dilation and length contraction, only $$\alpha$$ is correct and $$\beta$$ is not correct. They used $$\beta$$ to justify length contraction. Thing to be note down is that there are no two frames. Observer A is measuring time for event in moving frame, so light remains in moving frame. But this is ridiculous to think that speed remain same and not time, while it's clear that length is increased. So increased speed equalize time.

Also using of pythogoras theorem is not justifiable. Because event took place in moving frame of B, and he took his observation without concern of any relative speed. Thus the term with relative speed should added to observer A, at rest. But time is used of frame B, because this event in B is time like.

Proof: That actually $$\beta$$ should be correct answer for time dilation. Suppose event in moving frame with speed $$v$$, and it's a point observed for time $$t_1$$. Now observer at relatively rest observe time $$t_2$$, but for him, frame has relative speed. So events for moving and rest frame $$s_1$$ and $$s_2$$ are given by,$$s_1^2=c^2t_1^2\quad\text{and}\quad s_2^2=c^2t_2^2+v^2t_2^2$$As both observe same event thus there is no difference in event's measurement,$$s_2^2=s_1^2=t_1^2=\left(1+\frac{v^2}{c^2}\right)t_2^2\\\Rightarrow \frac{t_2^2}{t_1^2}=\frac{1}{1+\frac{v^2}{c^2}}=1-\frac{v^2}{c^2}\\\Rightarrow t_1=\frac{t_2}{\sqrt{1-\frac{v^2}{c^2}}}=\gamma t_2$$There is no requirement for separate time dilation in non-relative direction. Length contraction comes from time dilation because that is in direction of relative motion. In perpendicular direction, slope of length is inverse, so length is elongated in that direction.

https://physics.stackexchange.com/a/729615/344834

https://physics.stackexchange.com/a/733891/344834