Special theory of relativity - Lorentz transforms

Question

I have been reading about the STR & my first problem came very much in the beginning. So, as per the simplest form of the Lorentz transforms, we have

$$x' = (x - vt)\gamma \tag{1.1}$$

$$t' = (t - vx/c^2)\gamma \tag{1.2}$$

$$\gamma = \frac{1}{\sqrt{1- v^2/c^2}} \tag{1.3}$$

And this makes sense. However, then it proceeds to the length contraction & time dilation part. Where we get that

$$L = L'/\gamma \tag{2.1}$$

$$T = \gamma T' \tag{2.2}$$

I don't understand this. This seems to have a conflict with Lorentz transforms we stated above. Since in both of those transforms, the gamma was the denominator. Also, this relationship between length & time implies that they're inversely proportional, since $L*T$ cancels out the gamma, as opposed to the regular relationship between the space & time which is directly proportional. One more thing I find odd is that,

$$x' = ct' \tag{3}$$

Satisfies the speed of light in vaccuo constraint & so does the

$$x = ct \tag{4}$$

However, while, if we say that

$$c = L'/T' \tag{5.1}$$

$$c \ne L/T \tag{5.2}$$

Since,

$L/T = L'/(T'\gamma^2) \tag{6}$

$L/T$ will only be equal to $L'/T'$ if $v=0$ or very insignificant. But in that case, this function collapses to the simple Galleili transforms since the frame $K'$ would no longer be considered a moving frame but rather just a stationary frame shifted slightly at the x axis.

Where am I getting it wrong? Because, to me, the formulas presented for time dilation & length contraction seem in direct conflict with one another & also, if considered together seem in conflict with Lorentz transforms & speed of light in vaccuo.

Answer 1

Yes, this is tricky. But it's basic.

The Lorentz transformations apply to events, $(x,t)$ pairs, which are fundamental. Lengths and times - your $L$ and $T$ - are secondary constructs from events.

You measure the length of an object by aligning the ends against a ruler, reading the distances off against the scale, and subtracting them. If the ruler and the object are relatively at rest, then that's all there is to it. But if the object is moving along the ruler then (obviously) the two readings must be made at the same time.

So an $x$ on its own is not a length. A length is a difference between two events, $x_1-x_2$, for which $t_1=t_2$.

Then Lorentz says $x_1'-x_2'=\gamma(x_1-vt_1)-\gamma(x_2-vt_2)=\gamma L$ as $t_1=t_2$. But $t_2' \neq t_2'$ . So this $x_1'-x_2'$ is not a length - unless it's referring to the frame in which the object is at rest, for which it doesn't matter when you take the measurements. Calling the length in this frame $L_0$, the proper-length, we have $L=L_0/\gamma$.

The successive ticks of a moving clock occur at the same $x'$, as the clock is stationary in its own frame. You see the time interval between ticks as $T=t_1-t_2=\gamma(t_1'+vx_1'/c^2)-\gamma(t_2'+vx_2'/c^2)= \gamma(t_1'-t_2')=\gamma T'$ because the clock being stationary means $x_1'=x_2'$.

A length $L$ is not just a difference in $x$, a time $T$ is not just a difference in $t$.

Answer 2

I hope this space diagram from my book may clarify. The LT takes the point $\mathrm B(0,L)$ in the spacecraft frame to $(\gamma Lv,\gamma L)$ in the Earth frame, and it takes the point $\mathrm S (L,0)$ in the spacecraft frame to the point $(\gamma L,\gamma Lv)$ in the Earth frame. This does not give the coordinate length of the spacecraft, which is measured on the horizontal axis. The speed of light in both frames can be seen from the signal from B to S.

Answer 3

You're confusing two concepts here

1. Event
2. Value of c remain constant

I can do Lorentz transformation which can change $\sqrt{3}$ to $2$ so does this mean irrational numbers are same as rational number, of course not. Measurement of $L$ and $T$ are two different events. And when we say c is constant it means photon, wave traveling at c will have same speed in every other inertial reference frame.

Extra: the thought process/way you constructed c out of L and T is very close to the spirit of Lorentz invariants you just need 4-vectors to mathematical formulate your intuition.