Michelson-Morley Experiment

Quote from: fjr66 on July 21, 2019, 11:51:50 PM

If there is no medium, then it is not a wave. Wave equation must differentiable at every point. But if there is no medium, then there is discontinuity at the tip. It is impossible element of wave to bent without medium.

Electromagnetic waves don't need medium.

Increase of electrostatic field at one point generates magnetic field in the next point,
decrease makes that magnetic field decrease and then increaee in negative direction, and so on.

That magnetic field does the same to the third point, making electric field increase,
then decrease and then increase in opposite direction, then decrease again, and so on.

The new electric field induces the magnetic oscillation in the fourth point, magnetic induces electric in fifth,
and the chain continues without a medium.

Electric to magnetic to electric to magnetic and so forth.
Hence the name: Electromagnetic Wave.

If there is medium, then it interferes with the electric-magnetic-electric chain generation and changes the speed of the wave.

Maxwell formulate all his theory based on ether. 

Quote from: Macarios on July 22, 2019, 06:31:44 AM

Quote from: fjr66 on July 21, 2019, 11:51:50 PM

If there is no medium, then it is not a wave. Wave equation must differentiable at every point. But if there is no medium, then there is discontinuity at the tip. It is impossible element of wave to bent without medium.

Electromagnetic waves don't need medium.

Increase of electrostatic field at one point generates magnetic field in the next point,
decrease makes that magnetic field decrease and then increaee in negative direction, and so on.

That magnetic field does the same to the third point, making electric field increase,
then decrease and then increase in opposite direction, then decrease again, and so on.

The new electric field induces the magnetic oscillation in the fourth point, magnetic induces electric in fifth,
and the chain continues without a medium.

Electric to magnetic to electric to magnetic and so forth.
Hence the name: Electromagnetic Wave.

If there is medium, then it interferes with the electric-magnetic-electric chain generation and changes the speed of the wave.

Maxwell formulate all his theory based on ether. 

At his time they couldn't measure electrostatic fields or changes without material particles and they thought that material particles were necessary for electric fields.
That's why they tried to implement aether theoretically,.
If there are no known particles, there muct be something unknown there.

But the inability to measure was resolved and it was finally shown that the aether assumption was a dead end.
(I guess that something similar will happen with today's "dark matter" and "dark energy". )

Try reading https://en.wikipedia.org/wiki/Luminiferous_aether, it might help.

So light have a velocity not equal c in rotating frame (also constant permitivity and permeability).

It is correct that light will not necessarily have a velocity of c in a rotating reference frame, with it instead varying depending on the direction.
However I am unsure of if the permitivity and permeability are constant in those frames or if they vary with direction as well.

This means we must reformulate all Maxwell equation respect to rotating frame.

Or we can just use a non-rotating frame.

How about to make it simple. Let just say light travel at aether with velocity c, regardless it in rotating frame or not, then all the problem was solved.

I already explained that that doesn't solve anything.
The aether full of contradictions.
It needs to be solid to support transverse waves.
But it needs to be a fluid with 0 viscosity to allow objects to move through it with no resistance.
It needs to move relative to Earth to obtain the results of abberation, but needs to be stationary relative to Earth to obtain the results of MM and MMX.
More importantly, light needs to be a particle to explain all the particle nature of light.

The aether model simply doesn't work.

How about you make it simple. Treat a photon as a particle which, like all particles, does not require a medium to travel in and has wave nature.

It needs to move relative to Earth to obtain the results of abberation,

But stellar abberation contain wrong assumption. Look at above picture. Starlight come in direction AB with velocity c. And earth move in direction CB with velocity v. So for observer on earth starlight coming with velocity \sqrt{v^2 + c^2} that is larger than c.

You may said that we cannot use pythagorean rule in special relativity, maybe something like Einstein velocity addition is needed. But how can we determining angle  alpha (the angle at which the telescope must be tilted) if not using pythagorean rule.

If we make velocity v increased, then  angle alpha increased too. But this is also make velocity along AC larger than c, which is violating special relativity postulat.

So either special relativity is incorrect or stellar abberation is wrong.

And maybe in special relativity we must postulating  not just speed of light that constant for all observer but its direction too. How speed stay constant when direction changing, whereas direction determined by pythagorean rule.

Quote from: JackBlack on July 22, 2019, 02:02:17 PM

It needs to move relative to Earth to obtain the results of abberation,

But stellar abberation contain wrong assumption. Look at above picture. Starlight come in direction AB with velocity c. And earth move in direction CB with velocity v. So for observer on earth starlight coming with velocity \sqrt{v^2 + c^2} that is larger than c.

You may said that we cannot use pythagorean rule in special relativity, maybe something like Einstein velocity addition is needed. But how can we determining angle  alpha (the angle at which the telescope must be tilted) if not using pythagorean rule.

If we make velocity v increased, then  angle alpha increased too. But this is also make velocity along AC larger than c, which is violating special relativity postulat.

So either special relativity is incorrect or stellar abberation is wrong.

And maybe in special relativity we must postulating  not just speed of light that constant for all observer but its direction too. How speed stay constant when direction changing, whereas direction determined by pythagorean rule.

For observer moving parallel with the source, it is safely to assume velocity of light is the same for both of them. But how about observer moving in perpendicular direction with the source. Is speed of light still constant for him.

But stellar abberation contain wrong assumption.

Stellar aberration itself didn't. The ballistic and aether models that tried to explain it did.
Under both of those models, the speed of light needed to be greater.

You may said that we cannot use pythagorean rule in special relativity, maybe something like Einstein velocity addition is needed. But how can we determining angle  alpha (the angle at which the telescope must be tilted) if not using pythagorean rule.

You don't use Pythagoras to determine the angle at all.
But there are 3 main ways. One is to determine it all in the reference frame where Earth is moving.
This works quite similar to the diagram above, but the hypotenuse is not the path light is taking.
Light is travelling straight down while the telescope moves to the side.
You use basically the same approach as your triangle, but as I said, light doesn't go down the hypotenuse so it isn't travelling faster than c.
The angle alpha is the inverse tan of the velocity of Earth divided by the speed of light.
You can then either note that the velocity of Earth is quite small and thus approximate the angle for an observer on Earth to be the same, or you can use more complex formulas to transfer between the frames.

Another is to consider it in the reference frame of Earth, noting that the light must be travelling at a velocity of c, and then using the velocity of the reference frame relative to Earth to determine the angle, meaning we use inverse sin instead of inverse tan.

The other option is to use the proper relativistic formulas for relative velocities to find the velocity of the light (in 2D for this case, 3D in more general cases), relative to the observer on Earth (noting that as Earth is moving with a velocity of v in one direction relative to the frame, the frame is moving with a velocity of -v relative to the Earth, although that shouldn't cause any issues).
The horizontal component is easy. It just uses the commonly used formula for relativistic speeds:
(0+v)/(1+0*v/c^2)=v.
So the horizontal component of the velocity of light relative to Earth will be v.

Up next is the vertical component. This is more complex.
Instead of the normal:
(u+v)/(1+u*v/c^2)
you instead have:
sqrt(1-v^2/c^2)*u_v/(1+v*u_h/c^2).
So sticking in the values for our system you get:
sqrt(1-v^2/c^2)*c/(1+v*0/c^2)
=sqrt(c^2-v^2)

Rather than figuring out this directly, it is easier to figure out what the speed will be along the hypotenuse, i.e. the speed along the direction light is travelling)
That will be the square-root of the sum of squares, i.e.:
u=sqrt(v^2+(sqrt(c^2-v^2))^2.
u=c.
And then like before we use inverse sine.
Note that this works out exactly the same as before.

So the options:
Using sine we get an angle of 20.64076007".
Using tan we get 20.64075997"
So basically the same.

But importantly, with the proper relativistic approach, the speed of light remains c in any inertial reference frame.
Increasing the speed of Earth will not make light travel faster relative to Earth.

So aberration isn't a problem for relativity.

How speed stay constant when direction changing, whereas direction determined by pythagorean rule.

Direction is determined from the component velocities, as is speed, but in different ways.
The speed is done by Pythagoras, but the direction is done by trig.
The component velocities change, and thus the direction does, but the overall speed remains the same.

Quote from: JackBlack on July 22, 2019, 02:02:17 PM

It needs to move relative to Earth to obtain the results of abberation,

But stellar abberation contain wrong assumption. Look at above picture. Starlight come in direction AB with velocity c. And earth move in direction CB with velocity v. So for observer on earth starlight coming with velocity \sqrt{v^2 + c^2} that is larger than c.

You may said that we cannot use pythagorean rule in special relativity, maybe something like Einstein velocity addition is needed. But how can we determining angle  alpha (the angle at which the telescope must be tilted) if not using pythagorean rule.

If we make velocity v increased, then  angle alpha increased too. But this is also make velocity along AC larger than c, which is violating special relativity postulat.

So either special relativity is incorrect or stellar abberation is wrong.

And maybe in special relativity we must postulating  not just speed of light that constant for all observer but its direction too. How speed stay constant when direction changing, whereas direction determined by pythagorean rule.

You can’t add velocities like that for EM radiation.

The angle alpha is the inverse tan of the velocity of Earth divided by the speed of light.

You still used trigonometric law to determining angle alpha, if you consistent you will end up with the problem.
Remember in trigonometric we know that sin^2 alpha + cos^2 alpha = 1, where sin alpha = v/r, and cos alpha = c/r.
But where we get the value of r? If r = sqrt(v^2+c^2 ) then that law is valid, that is
(v^2 )/(v^2 + c^2) + (c^2)/(v^2 + c^2) = 1. But this means r = sqrt(v^2+c^2 ) > c, violating postulat of special relativity.

If we used r = c, so velocity of light still c in hypotenuse, we get
sin^2 alpha + cos^2 alpha = v^2/c^2 + c^2/c^2 > 1, which is violating trigonometric laws.

So either special relativity is incorrect  or trigonometric law can not be used in this case.

But the problem is how we determining angle alpha if not using trigonometric law?

Direction is determined from the component velocities, as is speed, but in different ways.
The speed is done by Pythagoras, but the direction is done by trig.

We can't separated trigonometric rule from pythagoras law. They mutually consistent.

Even if light travel along hypotenusa with velocity c, we get it projection to vertical line AB that is c cos alpha < c, which is violating special relativity postulat. So observer that at rest in line AB will measure speed of light less than c.

Quote from: fjr66 on July 24, 2019, 04:54:53 AM

Quote from: JackBlack on July 22, 2019, 02:02:17 PM

It needs to move relative to Earth to obtain the results of abberation,

But stellar abberation contain wrong assumption. Look at above picture. Starlight come in direction AB with velocity c. And earth move in direction CB with velocity v. So for observer on earth starlight coming with velocity \sqrt{v^2 + c^2} that is larger than c.

You may said that we cannot use pythagorean rule in special relativity, maybe something like Einstein velocity addition is needed. But how can we determining angle  alpha (the angle at which the telescope must be tilted) if not using pythagorean rule.

If we make velocity v increased, then  angle alpha increased too. But this is also make velocity along AC larger than c, which is violating special relativity postulat.

So either special relativity is incorrect or stellar abberation is wrong.

And maybe in special relativity we must postulating  not just speed of light that constant for all observer but its direction too. How speed stay constant when direction changing, whereas direction determined by pythagorean rule.

You can’t add velocities like that for EM radiation.

Look at the picture below, it is from wikipedia article about stellar abberation.
https://en.wikipedia.org/wiki/Aberration_(astronomy)



If earth move with velocity v relative to the star, then he will measure speed of light = u_x + v. But what if he at rest relative to the star, then he just measure velocity of light = u_x = c sin theta < c.

Special relativity never put any concern to velocity direction (and its projection to any plane), it is just about magnitude. Thats why it creates so many paradox that impossible to solved.

(v^2 )/(v^2 + c^2) + (c^2)/(v^2 + c^2) = 1. But this means r = sqrt(v^2+c^2 ) > c, violating postulat of special relativity.

No, it doesn't.
That is because in this reference frame light is not travelling along the hypotenuse. So the sqrt(v^2+c^2) does not correspond to any actual speed.
So we do not violate special relativity.

If we used r = c, so velocity of light still c in hypotenuse, we get

That you are now using a different reference frame and thus the velocity will be different.
Light will now be travelling down the hypotenuse with a speed of c. This will give it a horizontal component of v and a vertical component of sqrt(c^2-v^2).

So no violation of trig laws and no violation of relativity.

Stop just adding the velocities while ignoring relativity.

What you are doing is no better than claiming if you are moving towards the source of the light it has to be going faster than c. It is ignoring relativity.

There is no violation here.

Special relativity never put any concern to velocity direction (and its projection to any plane), it is just about magnitude. Thats why it creates so many paradox that impossible to solved.

No, the directionality is important as well. It is just when you have anything other than a straight line the math gets much harder so people like to focus on a straight line.

I provided what you need to do to deal with the directionality.

Directly below the section you quoted it provides the relativistic explanation and shows how the speed of light is kept constant.

Quote from: fjr66 on July 24, 2019, 07:00:40 PM

(v^2 )/(v^2 + c^2) + (c^2)/(v^2 + c^2) = 1. But this means r = sqrt(v^2+c^2 ) > c, violating postulat of special relativity.

No, it doesn't.
That is because in this reference frame light is not travelling along the hypotenuse. So the sqrt(v^2+c^2) does not correspond to any actual speed.
So we do not violate special relativity.

Quote from: fjr66 on July 24, 2019, 07:00:40 PM

If we used r = c, so velocity of light still c in hypotenuse, we get

That you are now using a different reference frame and thus the velocity will be different.
Light will now be travelling down the hypotenuse with a speed of c. This will give it a horizontal component of v and a vertical component of sqrt(c^2-v^2).

So no violation of trig laws and no violation of relativity.

Stop just adding the velocities while ignoring relativity.

What you are doing is no better than claiming if you are moving towards the source of the light it has to be going faster than c. It is ignoring relativity.

There is no violation here.

Quote from: fjr66 on July 24, 2019, 07:59:44 PM

Special relativity never put any concern to velocity direction (and its projection to any plane), it is just about magnitude. Thats why it creates so many paradox that impossible to solved.

No, the directionality is important as well. It is just when you have anything other than a straight line the math gets much harder so people like to focus on a straight line.

I provided what you need to do to deal with the directionality.

Directly below the section you quoted it provides the relativistic explanation and shows how the speed of light is kept constant.

If light travel at hypotenusa AC with velocity c. Then there is observer in horizontal line moving parallel with BC line, with velocity c sin alpha. For this observer light only have a vertical component parallel AB line  with velocity  c cos alpha which is less than c (also violating special relativity postulate).

If light travel at hypotenusa AC with velocity c. Then there is observer in horizontal line moving parallel with BC line, with velocity c sin alpha. For this observer light only have a vertical component parallel AB line  with velocity  c cos alpha which is less than c (also violating special relativity postulate).

Again, there is no violation.
Light travelling down the hypotenuse at a velocity of c is for the observer on Earth, where light has a vertical and horizontal component to its velocity.
These components are less than c, but the overall speed of light is still c.

The postulate of special relativity is that the speed is constant in any inertial reference frame, not that all the individual components of the velocity of light is c.

For the observer that Earth is moving relative to, light is not travelling down the hypotenuse and to find the speed you need to use the proper relativistic formulas.

Quote from: sokarul on July 24, 2019, 06:54:38 AM

Quote from: fjr66 on July 24, 2019, 04:54:53 AM

Quote from: JackBlack on July 22, 2019, 02:02:17 PM

It needs to move relative to Earth to obtain the results of abberation,

But stellar abberation contain wrong assumption. Look at above picture. Starlight come in direction AB with velocity c. And earth move in direction CB with velocity v. So for observer on earth starlight coming with velocity \sqrt{v^2 + c^2} that is larger than c.

You may said that we cannot use pythagorean rule in special relativity, maybe something like Einstein velocity addition is needed. But how can we determining angle  alpha (the angle at which the telescope must be tilted) if not using pythagorean rule.

If we make velocity v increased, then  angle alpha increased too. But this is also make velocity along AC larger than c, which is violating special relativity postulat.

So either special relativity is incorrect or stellar abberation is wrong.

And maybe in special relativity we must postulating  not just speed of light that constant for all observer but its direction too. How speed stay constant when direction changing, whereas direction determined by pythagorean rule.

You can’t add velocities like that for EM radiation.

Look at the picture below, it is from wikipedia article about stellar abberation.
https://en.wikipedia.org/wiki/Aberration_(astronomy)



If earth move with velocity v relative to the star, then he will measure speed of light = u_x + v. But what if he at rest relative to the star, then he just measure velocity of light = u_x = c sin theta < c.

Special relativity never put any concern to velocity direction (and its projection to any plane), it is just about magnitude. Thats why it creates so many paradox that impossible to solved.

It’s best not to use the classical explanation, but rather the correct relativistic explanation.

That is because in this reference frame light is not travelling along the hypotenuse. So the sqrt(v^2+c^2) does not correspond to any actual speed.
So we do not violate special relativity.

Look at this picture for inertial observer O:

It’s best not to use the classical explanation, but rather the correct relativistic explanation.

Can you give the correct explanation for angle alpha?

I’m sure the angle formula is correct.

Im referring to the speed of light.
The simplified formula is:
V+c=c not v+c>c.

The speed of light is the same for any reference frame.

I’m sure the angle formula is correct.

Im referring to the speed of light.
The simplified formula is:
V+c=c not v+c>c.

The speed of light is the same for any reference frame.

We talk about angle alpha. If you said alpha = arctan (v_y / v_x) then other trigonometric rules can be applied too.

So v_y = c * sin alpha < c (violating special relativity postulat).

But if you said v_y = c and v_x also = c, then arctan (c / c) = 45 degree. So the beam of light will arrive at angle 45 degree whatever initial angle is. And experimental data showing that this is not the case.

It’s best not to use the classical explanation, but rather the correct relativistic explanation.

Even in relativistic explanation, they still use u_y = c sin theta  which is less than c.

As you can see what you posted, c is always c. You can not have anything greater than c.

If you are getting a greater than c value for anything you are doing something wrong.

As you can see what you posted, c is always c. You can not have anything greater than c.

If you are getting a greater than c value for anything you are doing something wrong.

I'm not said its value greater than  c. But when light have speed less than c along specific direction, then it is also violating special relativity postulate.

Quote from: sokarul on July 25, 2019, 09:31:31 AM

As you can see what you posted, c is always c. You can not have anything greater than c.

If you are getting a greater than c value for anything you are doing something wrong.

I'm not said its value greater than  c. But when light have speed less than c along specific direction, then it is also violating special relativity postulate.

That never happens either.

Keep in mind that is for a vacuum.  When light enters a medium it will slow down.

Look at this picture for inertial observer O:

Is this the picture for the inertial observer O?
If so, they see the light moving at that angle, with a speed of c.
They do not just see the vertical component.

If this is actually for an outside observing that is also observing O moving, then they would see the light moving down with a speed of c, at that angle.
In order to determine what the observer O is seeing you need to use the proper relativistic formula to determine what they would be seeing.
Using the non-relativistic formula will produce incorrect results.

This is not a difficult concept to understand.
Why do you repeatedly use a non-relativistic formula when it is firmly established that relativistic effects will play a significant role?

Even in relativistic explanation, they still use u_y = c sin theta  which is less than c.

Only if the light is travelling at an angle in the initial reference frame, in which case it is just looking at the y component of the light's velocity, which should be less than c.
Do you understand that?
If light is travelling at an angle, its speed will be c, but the x, y and z components will not be c. Instead they will be some fraction of c based upon the angle that the light is travelling.
If light is travelling along the x axis, then the x component of its velocity will be c and the other components will be 0.

Again, this is not a difficult concept to understand.
Why pretend that this component of the velocity magically means light isn't travelling at c?

And also note again, this is the angle in the reference frame of the observer who is seeing Earth moving.
The component in the reference frame of Earth us u'_y this is u_y/gamma. This is c*sin(theta)/gamma.
This is not just c*sin(theta).

Quote from: fjr66 on July 25, 2019, 02:26:42 AM

Look at this picture for inertial observer O:

Is this the picture for the inertial observer O?
If so, they see the light moving at that angle, with a speed of c.
They do not just see the vertical component.

If this is actually for an outside observing that is also observing O moving, then they would see the light moving down with a speed of c, at that angle.
In order to determine what the observer O is seeing you need to use the proper relativistic formula to determine what they would be seeing.
Using the non-relativistic formula will produce incorrect results.

This is not a difficult concept to understand.
Why do you repeatedly use a non-relativistic formula when it is firmly established that relativistic effects will play a significant role?

Observer O never see light travel at angle, they just see it moving down  straight with velocity less than c. It is observer P that at rest with the source that see light moving with angle alpha with velocity c. It is like when you throw a ball inside a moving car, then you never see horizontal component of velocity of the ball. You just see it moving straight vertically. But what if for observer outside that standing in the road, they will see the ball moving at angle relative to vertical.

The problem is if observer O and P measuring the same speed of light like what  postulated by special relativity, then how we can determining the correct value for angle alpha. If for observer P see light travel at  hypotenusa with velocity c, and observer O see light travel down also with velocity c, then that means cos alpha = c/c. So alpha always  = 90 degree, whatever angle alpha in reality is.

Quote from: fjr66 on July 25, 2019, 10:05:21 AM

Quote from: sokarul on July 25, 2019, 09:31:31 AM

As you can see what you posted, c is always c. You can not have anything greater than c.

If you are getting a greater than c value for anything you are doing something wrong.

I'm not said its value greater than  c. But when light have speed less than c along specific direction, then it is also violating special relativity postulate.

That never happens either.

Keep in mind that is for a vacuum.  When light enters a medium it will slow down.

I never talk about light travel at vaccum. I talk about velocity projection. Is that have the same magnitudo with the velocity itself?

Only if the light is travelling at an angle in the initial reference frame, in which case it is just looking at the y component of the light's velocity, which should be less than c.
Do you understand that?

Then if there is any observer that co moving with one velocity component, they will see light travel with velocity less than c. Like what I said before. For outside observer that stand in the road, they see the ball travel at angle with velocity v and have velocity component v_x and v_y which its value less than v. But what is for observer in the car that moving with velocity v = v_x? They never see the ball have velocity component v_x, they just see it moving vertically with velocity less than v that is v_y.

Quote from: JackBlack on July 25, 2019, 03:58:36 PM

Quote from: fjr66 on July 25, 2019, 02:26:42 AM

Look at this picture for inertial observer O:

Is this the picture for the inertial observer O?
If so, they see the light moving at that angle, with a speed of c.
They do not just see the vertical component.

If this is actually for an outside observing that is also observing O moving, then they would see the light moving down with a speed of c, at that angle.
In order to determine what the observer O is seeing you need to use the proper relativistic formula to determine what they would be seeing.
Using the non-relativistic formula will produce incorrect results.

This is not a difficult concept to understand.
Why do you repeatedly use a non-relativistic formula when it is firmly established that relativistic effects will play a significant role?

Observer O never see light travel at angle, they just see it moving down  straight with velocity less than c. It is observer P that at rest with the source that see light moving with angle alpha with velocity c. It is like when you throw a ball inside a moving car, then you never see horizontal component of velocity of the ball. You just see it moving straight vertically. But what if for observer outside that standing in the road, they will see the ball moving at angle relative to vertical.

The problem is if observer O and P measuring the same speed of light like what  postulated by special relativity, then how we can determining the correct value for angle alpha. If for observer P see light travel at  hypotenusa with velocity c, and observer O see light travel down also with velocity c, then that means cos alpha = c/c. So alpha always  = 90 degree, whatever angle alpha in reality is.

Each observer sees the light with a different frequency.

Quote from: sokarul on July 25, 2019, 02:07:46 PM

Quote from: fjr66 on July 25, 2019, 10:05:21 AM

Quote from: sokarul on July 25, 2019, 09:31:31 AM

As you can see what you posted, c is always c. You can not have anything greater than c.

If you are getting a greater than c value for anything you are doing something wrong.

I'm not said its value greater than  c. But when light have speed less than c along specific direction, then it is also violating special relativity postulate.

That never happens either.

Keep in mind that is for a vacuum.  When light enters a medium it will slow down.

I never talk about light travel at vaccum. I talk about velocity projection. Is that have the same magnitudo with the velocity itself?

As the link you quoted says, c is constant.

As the link you quoted says, c is constant.

So tan alpha = c/c = 1
cos alpha also  = c/c = 1
and sin alpha also = c/c = 1

then this will destroying trigonometric law.
Impossible we can determined what angle alpha is.

Each observer sees the light with a different frequency.

I never talk about frequency or Doppler Shift. I talk about the correct way to determining angle alpha.

Quote from: JackBlack on July 22, 2019, 02:02:17 PM

It needs to move relative to Earth to obtain the results of abberation,

But stellar abberation contain wrong assumption. Look at above picture. Starlight come in direction AB with velocity c. And earth move in direction CB with velocity v. So for observer on earth starlight coming with velocity \sqrt{v^2 + c^2} that is larger than c.

You may said that we cannot use pythagorean rule in special relativity, maybe something like Einstein velocity addition is needed. But how can we determining angle  alpha (the angle at which the telescope must be tilted) if not using pythagorean rule.

If we make velocity v increased, then  angle alpha increased too. But this is also make velocity along AC larger than c, which is violating special relativity postulat.

So either special relativity is incorrect or stellar abberation is wrong.

Or you have ignored time dilation.

And maybe in special relativity we must postulating  not just speed of light that constant for all observer but its direction too. How speed stay constant when direction changing, whereas direction determined by pythagorean rule.