Lorentz contraction in Special Relativity

[AA Institute]

Does the equation for Lorentz contraction stay exactly the same when
quantifying the 'mass increase factor' at relativistic speeds? Further,
when one moves at close to light speeds, the view of the surrounding
universe seen from the ship is distorted into a circular window. Would
the radius of such a window contract in line with the same equations of
Lorentz contraction?


AA
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[mitchellmckain]

The first question does not make much sense to me. There is no mass in the lorentz contraction equations. This distortion into a circular window sounds a bit like the aberration of light and the calculation for this already takes the lorentz contraction into account.

 

[AA Institute]

From this page:

http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

The relativistic mass increases according to the equation:

mr = m0 /sqrt(1 - v2/c2)

So I want to know, is the equation for Lorentz contraction *exactly* the same as this? To put this another way, the apparent relativistic contraction of an object approaching a velocity close to 'c' must be governed by an equation. Is that equation the same as the one I stated above (for 'mass increase') or is it something different?

In my second question, I wanted to know if the radius of the distortion into a circular window due to the aberration of light, could be calculated (and was governed) by the same equation as Lorentz contraction? Or would that be a different set of equations altogether?

Consider a starship cruising through deep space at speeds approaching the speed of light. When traveling at *exactly* the speed of light, would the circular window in front of such a starship have contracted to a single point of light? So that from the perspective of an observer situated on such a light-speed ship, the entire universe's surrounding starlight will have shrunk to a single point source?

Thanx for further advice,

AA
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[Doc Al]

The relativistic mass increases according to the equation:

mr = m0 /sqrt(1 - v2/c2)

So I want to know, is the equation for Lorentz contraction *exactly* the same as this? To put this another way, the apparent relativistic contraction of an object approaching a velocity close to 'c' must be governed by an equation. Is that equation the same as the one I stated above (for 'mass increase') or is it something different?

It had better be somewhat different, since "relativistic" mass increases with speed, while measured lengths decrease with speed.
$L = L_0 \sqrt{1 - v^2/c^2}$ gives the measured length of an object along its direction of motion.

 

[BioBen]

So I want to know, is the equation for Lorentz contraction *exactly* the same as this?

The equation is nearly the same (same form) :
L = L0*sqrt(1-v²/c²)

for 'mass increase'

I don't know how you call it in english, but i think it is better to say that mass is a relativistic invariant (mass = m0), what changes is intertia, which is defined as m0/sqrt(1-v²/c²).

Benjamin

 

[JesseM]

Does the equation for Lorentz contraction stay exactly the same when
quantifying the 'mass increase factor' at relativistic speeds? Further,
when one moves at close to light speeds, the view of the surrounding
universe seen from the ship is distorted into a circular window. Would
the radius of such a window contract in line with the same equations of
Lorentz contraction?

This comment by jtbell from another thread, on the difference between what someone "observes" vs. what he "sees" in relativity, may be helpful here:

In relativity, we must distinguish between what a single observer "sees" (using the light rays that travel from an object to his eyes) and what he "observes" (after correcting for the time that it takes for those rays to travel from the object to him).

When we talk about length contraction etc., by default we talk about "observing" the object either as described above; or by having several observers with synchronized clocks, and using a common spatial coordinate system, each located at key points, or close enough to them that the light-propagation time is negligible. For example, to measure the length of a moving rod, we could have two observers at rest with respect to each other, position themselves such that the two ends of the rod pass next to them simultaneously, in the inertial reference frame in which the observers are at rest.

When someone wants to describe the actual visual appearance of relativistically moving objects, he usually says very explicitly that that's what he's doing.

Lorentz contraction falls into the category of what is observed after you correct for delays due to the speed of light, while the outside world being distorted into a "circular window" falls into the category of what is seen without any such corrections.

 

[DrGreg]

The relativistic mass increases according to the equation:

mr = m0 /sqrt(1 - v2/c2)

So I want to know, is the equation for Lorentz contraction *exactly* the same as this?

The Lorentz equations for distance and time are more than just "length contraction" and "time dilation".

In full they are:

$$t_2 = \gamma (t_1 - v \ x_1 / c^2)$$
$$x_2 = \gamma (x_1 - v \ t_1)$$

where $\gamma = 1 / \sqrt{1 - v^2 / c^2}$ and v is the velocity of observer 2 (in the direction of x) relative to observer 1.

If you do not know these equations you should take some time to think about them and work out what they really mean.

It turns out that relativistic mass m_r and relativistic momentum p behave like time and space. The Lorentz equations in this case are

$$m_{r2} = \gamma (m_{r1} - p_1 \ v / c^2)$$
$$p_2 = \gamma (p_1 - m_{r1} \ v)$$

Remember v is the velocity of observer 2 relative to observer 1 and not the velocity of the particle whose mass & momentum you are measuring.

However if the particle is stationary relative to observer 1, then m_r1 = m₀ and p₁ = 0, so

$$m_{r2} = \gamma m_{0}$$
$$p_2 = - \gamma m_{0} \ v$$

 

[AA Institute]

Okay, I am beginning to get the picture with all the different viewpoints in Lorentz Contraction and the various field equations that one can work with to give different perspectives.

But ultimately, what I am looking for is the formula or rule for computing the apparent angular width (diameter?) of the relativistic window that an observer onboard a near-lightspeed ship will see squeezed in front of him, as his vehicle upspeeds toward 'c'.

The universe's total surrounding starlight will be seen to get squeezed into a narrow field of view in front of the observer, so at what rate does the compression of that window happen, relative to 'c'?

Something like this, perhaps:

at 99.9% c, window width =120 degrees
at 99.99% c, window width =45 degrees
at 99.999% c, window width =2.5 degrees
at 99.9999% c, window width =0.05 degrees
at 100% c, window width =0.0 degrees

(these numbers are made up :) of course!)

 

[robphy]

Are you referring to the "headlight effect"?
e.g., http://physics.tamuk.edu/~hewett/Mo...locities/HeadlightEffect/HeadlightEffect.html

 

[Jimmy Snyder]

When traveling at *exactly* the speed of light, would the circular window in front of such a starship have contracted to a single point of light?

The question is unphysical because massive objects cannot travel at that speed. However, it may be instructive to consider what a sentient photon would see. After all, Einstein spoke of imagining what it would be like to travel along side a beam of light. I don't know what images he evoked.

What follows are my own fevered imaginings:

As particles approach the speed of light, we see their clocks slow down. Extrapolating to the speed of light, we should 'see' a photon's clock stopped. That is, from our point of view, time does not pass for a photon and so it cannot accumulate experiences. It doesn't see anything at all.

 

[Eridanus1]

I thought the universe starlight condensed to a point... Like what Abdul Ahad's constant was saying:)

http://siliconvirtue.com/read.php?f=141&i=839&t=826

and bottom of this page:)

http://www.astroscience.org/abdul-ahad/aa-glossary.htm

E

 

[Jimmy Snyder]

I thought the universe starlight condensed to a point... Like what Abdul Ahad's constant was saying:)

http://siliconvirtue.com/read.php?f=141&i=839&t=826

and bottom of this page:)

http://www.astroscience.org/abdul-ahad/aa-glossary.htm

E

In the next to last paragraph of the second link, I found this:

Further, the convergence of the entire universe into a circular dot of light (figure 4 above), whose dimensions are zero and whose flux amounts to Ahad's constant of -6.5 magnitudes, will only be possible at a theoretical ship velocity of exactly 'c'.

Since an actual spaceship cannot achieve that velocity, he is saying that the convergence to a point is not possible.

 

[Trilairian]

Does the equation for Lorentz contraction stay exactly the same when
quantifying the 'mass increase factor' at relativistic speeds? Further,
when one moves at close to light speeds, the view of the surrounding
universe seen from the ship is distorted into a circular window. Would
the radius of such a window contract in line with the same equations of
Lorentz contraction?


AA
------------------------------**--------------------------
http://www.publishedauthors.net/aa_spaceagent/
"The ultimate dream adventure awaiting humanity..."
------------------------------**--------------------------

Mass doesn't change with speed.