# The Lorentz Transformations and vector components

• anantchowdhary
In summary, the Lorentz transformation does not always hold true for length contraction. The correct formula is L_0 = \frac {L} { \sqrt {1 - \frac {{v_x}^2} {c^2}}}

#### anantchowdhary

If a rod is traveling with a velocity 'v' and its proper length is $$L_0$$

will the lorentz transformations given below hold true for the length contraction

$$L_0 = \frac {L} { \sqrt {1 - \frac {{v_x}^2} {c^2}}}$$

$$L_0 = \frac {L} { \sqrt {1 - \frac {{v_y}^2} {c^2}}}$$

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No, that is incorrect, first you apply the Gamma transform in one dimension. Second your formula (did you try it?) does not contract, it expands.

Let v= .9c what does your formula predict for the length?

OH! pardon me, did you really want to express $L_0$ as a function of v ? Or did you mean to write:

$$L =\frac {L_0} \sqrt {1 - {( \frac v c )^2}}$$

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Err...i meant to write L to be the Length as observed by an observer who is observing the moving rod

and if the rod is going at an angle relative to the x-axis or to the y axis,then wouldn't we first calculate the contraction of its 'x' component and then see the new length?

In special relativity the motion is assumed to be aligned with the x axis. The Lorentz transform of the y-axis is

y= y'

Integral said:
No, that is incorrect, first you apply the Gamma transform in one dimension. Second your formula (did you try it?) does not contract, it expands.

Let v= .9c what does your formula predict for the length?

OH! pardon me, did you really want to express $L_0$ as a function of v ? Or did you mean to write:

$$L =\frac {L_0} \sqrt {1 - {( \frac v c )^2}}$$

Opps! I kinda messed that up, this why I do not like to work off the top of my head. (I do not have a reference handy)

$$L = {L_0} \sqrt {1 - {( \frac v c )^2}}$$

There, now we have contraction!

ok.But what of we hav a rod traveling at an angle?How would we calculate the length conrtraction without taking its components?

Integral said:
In special relativity the motion is assumed to be aligned with the x axis. The Lorentz transform of the y-axis is

y= y'

That is something we use for simplicity, isn't it? It makes deriving formulae easier.

Since in Special Relativity, motion is linear, you can always align the axis with the motion. Why wouldn't you?

i agree with neutrino.But why isn't the calculation of the contraction taking components correct?

for example,as u said if we assume the motion to be in the direction of the x axis,then we can simply say the relative velocity along the y-axis is zero

Therefore
$$L_0_y = \frac {L_y} { \sqrt {1 - \frac {{0}^2} {c^2}}}$$

would giv us $$L_0_y=L_y$$

therefore there isn't any contraction in the direction of the 'y' axis!

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anantchowdhary said:
i agree with neutrino.But why isn't the calculation of the contraction taking components correct?

for example,as u said if we assume the motion to be in the direction of the x axis,then we can simply say the relative velocity along the y-axis is zero

Therefore
$$L_0 = \frac {L} { \sqrt {1 - \frac {{0}^2} {c^2}}}$$

would giv us $$L_0_y=L_y$$

therefore there isn't any contraction in the direction of the 'y' axis!

Your equation, as written, is pretty useless. It seems to imply that L 0 is a function of L. Do you really want to say that L 0 is changing?

An object only contracts in the direction of motion. That is a simple fact, perhaps you need to look a bit more carefully at the FULL set of Lorentz transforms.

PLease tell me then,how would we calculate the length contraction if the object is moving with an angle with respect to the horizontal?

Once again, align your coordinate system with the direction of linear motion.

ok,ill ask a question related to the topic:

If we hav say an equilateral trinagle(a cardboard cutout) and its traveling with a certain velocity with respect to a person A.Would he observer the angles to be any different from 60 degrees?(when compared to the proper angles)

Integral said:
Since in Special Relativity, motion is linear, you can always align the axis with the motion. Why wouldn't you?
Obviously for three or more objects with different velocities, except for trivial cases, that would not be the case.

In such cases you can use the velocity addition formula.
Note, however, that this formula is neither commutative or associative except for trivial cases!

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a simpler problem but instructive

anantchowdhary said:
ok,ill ask a question related to the topic:

If we hav say an equilateral trinagle(a cardboard cutout) and its traveling with a certain velocity with respect to a person A.Would he observer the angles to be any different from 60 degrees?(when compared to the proper angles)
I used to solve with myh students the follolwing problem. Consider a rod of given proper length at rest in I. It makes a given angle (theta') with the positive direction of the O'X' axis. Find out its length measured by observers of the I frame relative to which it moves with V.

Umm...according to me,the radius of the rod(cylindrical) should change...correct?Are u a teacher?

bernhard.rothenstein said:
I used to solve with myh students the follolwing problem. Consider a rod of given proper length at rest in I. It makes a given angle (theta') with the positive direction of the O'X' axis. Find out its length measured by observers of the I frame relative to which it moves with V.

I thought it was at rest w.r.t I?

Or did you mean to say "length measured by observers of the I' frame relative to which it moves with V." If that's the case, will you accept this soltuion?

$$L_{x} = \frac{L_{0x}}{\gamma\left(v_{cm}\right)} = \frac{L_{0}\cos{\theta}}{\gamma\left(v_{cm}\right)}$$

$$L_{y} = \frac{L_{0y}}{\gamma\left(v_{cm}\right)} = \frac{L_{0}\sin{\theta}}{\gamma\left(v_{cm}\right)}$$

$$\therefore L = \sqrt{L_{x}^{2} + L_{y}^{2}} = \frac{L_0}{\gamma}$$

Edit: After reading this post by jtbell, I think I might've made a mistake.

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I don't think uve made a mistake!I think only the radius of the cylinder should change,as the direction of motion of the rod is in line with the x-axis only the rod is at an angle but traveling at the same alignment

neutrino's analysis is in error; as jtbell explains, only the component of length parallel to the direction of motion will contract.
anantchowdhary said:
I don't think uve made a mistake!I think only the radius of the cylinder should change,as the direction of motion of the rod is in line with the x-axis only the rod is at an angle but traveling at the same alignment
What are you talking about with the radius of the cylinder? Assume it's a thin rod. A moving observer will measure both the length and the angle of the rod to be different, compared to observations made in the stick's proper frame.

err...i still can't get this.Why would the angle and length be different.The rod is still traveling parallel to the x axis.Even though it is at an angle,its motion is not at an angle with the x-axis isn't it?

Doc Al said:
neutrino's analysis is in error; as jtbell explains,
Why is there an error?Also,if I've understood the question correctly, the rod isn't a vector!Its velocity is a vector which is just parallel to the x axis.

Thank you

the oblique rod poblem

anantchowdhary said:
Why is there an error?Also,if I've understood the question correctly, the rod isn't a vector!Its velocity is a vector which is just parallel to the x axis.

Thank you
In my statement of the problem consider please that the rod (not a cylinder) is at rest in I'. Take into account that distances measured perpendicular to the direction of relative motion are invariants and that the OX(O'X') components contract
L(x)=L(0)cosw'/g(V)
L(y)=L(0)sinw'
where L(0) stands for the proper length, w' for the angle made by the rod with the positive direction of the overlaped axes and g(V) is the gamma factor. In order to simplify the problem consider that one of the ends of the rod is located at the origin O' of the I' frame. The fact that you consider the rod as a vector does not change the problem.
In order to finish apply Pythagoras' theorem. As an interesting fact notice that when detected from I the rod is rotated and you can find out the angle w measured by observers from I
tgw=L(y)/L(x).
Expressing the right side of that equation as a function of physical quantities measured in I' we obtain the relationship between w and w'.
I used to ask my students capisci?

anantchowdhary said:
Why is there an error?Also,if I've understood the question correctly, the rod isn't a vector!Its velocity is a vector which is just parallel to the x axis.

You will agree that any object has dimensions parallel and perpendicular to its relative velocity vector, right? In this case, the velocity is parallel to the x-axis, so all length measurements parallel to the x-axis will be contracted. (And length measurements perpendicular to the x-axis will remain unchanged.)

Rather than discuss this same issue in multiple threads, followups should go here: https://www.physicsforums.com/showthread.php?t=158375". (Especially since jtbell has given the explicit calculation in that thread.)

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## 1. What are the Lorentz Transformations?

The Lorentz Transformations are a set of mathematical equations used in the theory of special relativity to describe how measurements of space and time are affected by the relative motion of two observers.

## 2. How do the Lorentz Transformations affect vector components?

The Lorentz Transformations establish a relationship between the coordinates of an event measured by two observers in relative motion. This affects the vector components by showing that they are not invariant under transformations between reference frames.

## 3. What is the significance of the Lorentz factor in the transformations?

The Lorentz factor (γ) is a key component in the Lorentz Transformations as it accounts for the time dilation and length contraction effects predicted by special relativity. It is also used to calculate the velocity of an object in different reference frames.

## 4. How do the Lorentz Transformations relate to Einstein's theory of special relativity?

Einstein's theory of special relativity is based on the idea that the laws of physics are the same for all observers in uniform motion. The Lorentz Transformations provide the mathematical framework for this theory, allowing for the prediction and explanation of phenomena such as time dilation and length contraction.

## 5. Can the Lorentz Transformations be applied to objects moving at speeds close to the speed of light?

Yes, the Lorentz Transformations can be applied to objects moving at any speed, including those close to the speed of light. In fact, they were originally developed to describe the behavior of objects moving at high speeds, and have been confirmed by numerous experiments and observations.