Thanks Harry, Tom & Shevek, I studied your posts carefully.

shevek wrote:
Tom Roberts wrote:
Ken S. Tucker wrote:
Tom Roberts wrote:
No. It just requires less force to accelerate a moving particle
transverse to its motion compared to the force needed for the same
acceleration along its motion (here all quantities are referred to a
single inertial frame).

That's pontification galore, care to underwrite
that statement with real dynamics?

The equation of motion in Minkowski spacetime for a pointlike particle
of fixed mass m under the influence of 4-force F is:
F = dP/d\tau
P = m U
where P is the particle's 4-momentum, U is its 4-velocity, and \tau is
its proper time.

Project this equation onto inertial coordinates {x,y,z,t} in which the
particle is moving along the z axis with velocity v, and the above
equation becomes:
F^x = \gamma m d^2x(t)/dt^2
F^y = \gamma m d^2y(t)/dt^2
F^z = \gamma^3 m d^2z(t)/dt^2
\gamma = 1/sqrt(1-v^2)
here {x(t),y(t),z(t),t} is the trajectory of the particle.

As \gamma 1, it clearly requires more force for a given acceleration
along the z axis than it does along either the x or y axes.

Exercise for the reader: in those coordinates there is a 4th
equation which I omitted. What is it? Why did I omit it?



Thanks for the exercise Tom!

The other equation is

F^t = \gamma/c * dE/dt = \gamma/c * (F dot u)

Where E is energy = m_0 \gamma c^2.

I think you omitted it because this component gives information about
the energy rather than the trajectory. Energy is conserved provided
the force is always normal to the motion, as in the case of a charged
particle in a pure magnetic field.

This exercise can be done perhaps more simply without defining a proper
time \tau by defining momentum in terms of the rest mass:

p = \gamma * m_0 * u

and then applying the product rule when taking the derivative:

F = dp/dt
F = \gamma * m_0 * du/dt + m_0 * u * d\gamma/dt

The second term above is in the direction of the velocity, and so thus
only contributes to the force component parallel to the velocity.

Cheers - shevek

Tom's physics comes from 1905 SR (see Principle of Relativty pg 63).

While it would be fun to use the GR

dU^i/ds = GAMMA^i _uv U^u U^v

geodesical equation, I'd suggest a simpler scenario.

Using a laser emitted from some fixed "inertial frame" accelerate
an object and predict it's rate of acceleration in the inertial frame.
I think that's a simple and reasonable test gedanken.

My concern is a Doppler shift that does not appear in any of the
equations set forth so far. Also does the transverse acceleration
account for aberration.

I was hoping to others realist dynamics but keep it as simple as
possible.

TIA
Ken

shevek wrote:
Thanks for the exercise Tom!

The other equation is

F^t = \gamma/c * dE/dt = \gamma/c * (F dot u)

Where E is energy = m_0 \gamma c^2.

I think you omitted it because this component gives information about
the energy rather than the trajectory. Energy is conserved provided
the force is always normal to the motion, as in the case of a charged
particle in a pure magnetic field.

This exercise can be done perhaps more simply without defining a proper
time \tau by defining momentum in terms of the rest mass:

p = \gamma * m_0 * u

and then applying the product rule when taking the derivative:

F = dp/dt
F = \gamma * m_0 * du/dt + m_0 * u * d\gamma/dt

The second term above is in the direction of the velocity, and so thus
only contributes to the force component parallel to the velocity.

If a missile was propelled through an ultramundane aether which offered
resistance by a factor of gamma and a booster rocket was positioned on
the side of the missile, would the energy needed in the booster rocket
to change the direction of the missile be less than that in the main
rocket propelling the missile in the original direction? I would say
yes because there is less aether resistance normal to the original
direction of the missile. However, it is a bit more complicated as the
motion of the missile through the aether can have a direct analogy to
the motion of an object through the air. In such cases the excess
pressure from the object moving forward is dissipated spherically in
waves, right? Isn't that what causes the resistance? Does the math
really offer an explanation of why it takes less energy to move
tangentially rather than parallel? I think it has to be explained in
terms of a model. In a vacuum there would be no reason for the
resistance.

Vern

wrote:
shevek wrote:
Thanks for the exercise Tom!

The other equation is

F^t = \gamma/c * dE/dt = \gamma/c * (F dot u)

Where E is energy = m_0 \gamma c^2.

I think you omitted it because this component gives information about
the energy rather than the trajectory. Energy is conserved provided
the force is always normal to the motion, as in the case of a charged
particle in a pure magnetic field.

This exercise can be done perhaps more simply without defining a proper
time \tau by defining momentum in terms of the rest mass:

p = \gamma * m_0 * u

and then applying the product rule when taking the derivative:

F = dp/dt
F = \gamma * m_0 * du/dt + m_0 * u * d\gamma/dt

The second term above is in the direction of the velocity, and so thus
only contributes to the force component parallel to the velocity.

If a missile was propelled through an ultramundane aether which offered
resistance by a factor of gamma and a booster rocket was positioned on
the side of the missile, would the energy needed in the booster rocket
to change the direction of the missile be less than that in the main
rocket propelling the missile in the original direction?

Yes, that's what the equations above are telling us. However, if you
move to the frame of the missile, you will see that the energy needed
for acceleration (as measured in that frame) is the same in any
direction. The affect in question (more energy needed to accelerate
longitudal motion than transverse) is a coordinate-dependent effect.


Incidentally, if the missile was at rest in the ultramundane aether,
and the laboratory was flying by, the observed effect would be the
same. The rest frame of this unltramundane aether is non needed in the
equation - hence this ultramundane aether is also superfluous as
pointed out by Einstein.

I would say
yes because there is less aether resistance normal to the original
direction of the missile. However, it is a bit more complicated as the
motion of the missile through the aether can have a direct analogy to
the motion of an object through the air.

True again. The effect is not a result of "resistance" in analogy with
air restistance. There is no such thing, just as there is no air
resistance for a phonon moving across the room. Rather, the effect is
due to the changes in our coordinate system, which is built on light
propagation and the principles of relativity.

In such cases the excess
pressure from the object moving forward is dissipated spherically in
waves, right? Isn't that what causes the resistance?

You were right, there's no direct analogy to the motion of an object
through air. That kind of aether theory has been disproven.

Does the math
really offer an explanation of why it takes less energy to move
tangentially rather than parallel? I think it has to be explained in
terms of a model. In a vacuum there would be no reason for the
resistance.


That depends on your model of a vacuum, doesn't it.

If you explain a vacuum as a kinetic distribution of space-time atoms,
then describe light waves in this medium, and then build a physical
coordinate system using these light waves, you should be able to
expalain the different energies required for the accelerations.

Thanks Vern - shevek

Vern

Ken S. Tucker wrote:
Thanks Harry, Tom & Shevek, I studied your posts carefully.

shevek wrote:
Tom Roberts wrote:
Ken S. Tucker wrote:
Tom Roberts wrote:
[..]
Tom's physics comes from 1905 SR (see Principle of Relativty pg 63).

While it would be fun to use the GR

dU^i/ds = GAMMA^i _uv U^u U^v

geodesical equation, I'd suggest a simpler scenario.

Using a laser emitted from some fixed "inertial frame" accelerate
an object and predict it's rate of acceleration in the inertial frame.
I think that's a simple and reasonable test gedanken.

My concern is a Doppler shift that does not appear in any of the
equations set forth so far.

Hi Ken-

You're right, that scenario does add more complexity. In that case,
the forces on the object - one from a laboritory laser aimed parallel
to the motion and one from a laboritory laser aimed perpendicular to
the motion, are not equal - due to the Doppler shift as you pointed
out. As the object aproaches the speed of light the longitudal laser
light will be redshifted and provide less and less force.

The laser will apply less and less force as the object accelerates due
to the doppler effect, and in addition that force will make less and
less acceleration due to the relativistic effects described. For your
example the two effects (doppler + relativistic coords) combine to make
the asymmetry even greater.

Cheers - shevek

shevek wrote:
yes because there is less aether resistance normal to the original
direction of the missile. However, it is a bit more complicated as the
motion of the missile through the aether can have a direct analogy to
the motion of an object through the air.

True again. The effect is not a result of "resistance" in analogy with
air restistance. There is no such thing, just as there is no air
resistance for a phonon moving across the room. Rather, the effect is
due to the changes in our coordinate system, which is built on light
propagation and the principles of relativity.

It's called viscoelasticity. With matter waves, they are perturbing
virtual particles when being moved.

In such cases the excess
pressure from the object moving forward is dissipated spherically in
waves, right? Isn't that what causes the resistance?

You were right, there's no direct analogy to the motion of an object
through air. That kind of aether theory has been disproven.

It's called terminal velocity. With matter waves, some of the
radiation is regenerated but they can only approach the driving speed.
The æther theory has not been disproven.

-Aut

Ken, stop misspelling its, you retard.

shevek:
laboritory - laboratory

shevek wrote:
wrote:

snip

If a missile was propelled through an ultramundane aether which offered
resistance by a factor of gamma and a booster rocket was positioned on
the side of the missile, would the energy needed in the booster rocket
to change the direction of the missile be less than that in the main
rocket propelling the missile in the original direction?

Yes, that's what the equations above are telling us. However, if you
move to the frame of the missile, you will see that the energy needed
for acceleration (as measured in that frame) is the same in any
direction. The affect in question (more energy needed to accelerate
longitudal motion than transverse) is a coordinate-dependent effect.

Right, but forget the equations for a second and consider the
assumption of an ultramundane aether. In the frame where the aether is
isotropic, the missile is accelerating. Since more aether resistance
is encountered in the direction of the motion, more energy would be
needed to accelerate the object in this frame. If you were to
accelerate the object normal to the direction of motion, from the
aether frame, you would only be creating a small aether resistance as
the velocity w.r.t the aether in the normal direction is not nearly as
much as the velocity w.r.t to the aether in the original direction. In
the frame of missile, you would still find that forces were acting on
the missile, a greater force coming from one direction (the direction
in which in the aether frame the missile is moving) and a smaller force
from normal to that direction (in which in the aether frame you were
attempting to accelerate normal to the original direction). So, with
the assumption of an aether, the effect is not frame dependent, it just
changes from resistance against a moving object to an unknown force (or
an aether wind) acting on a stationary object.

Incidentally, if the missile was at rest in the ultramundane aether,
and the laboratory was flying by, the observed effect would be the
same. The rest frame of this unltramundane aether is non needed in the
equation - hence this ultramundane aether is also superfluous as
pointed out by Einstein.

I'm suggesting a modification of Newton's laws to add in the effect
of aether resistance and I'm suggesting that without it there is no
explanation for the fact that it takes more force to accelerate an
object the faster you go and to also explain why it doesn't take as
much force to accelerate the object normal to its motion.

I would say
yes because there is less aether resistance normal to the original
direction of the missile. However, it is a bit more complicated as the
motion of the missile through the aether can have a direct analogy to
the motion of an object through the air.

True again. The effect is not a result of "resistance" in analogy with
air restistance. There is no such thing, just as there is no air
resistance for a phonon moving across the room. Rather, the effect is
due to the changes in our coordinate system, which is built on light
propagation and the principles of relativity.

In this thread we're not talking about a wave, photon or phonon's
resistance due to motion w.r.t a medium, but instead we are considering
an object's resistance to acceleration. With the assumption of an
ultramundane aether, the effect is not due to the changes in the
coordinate system.

In such cases the excess
pressure from the object moving forward is dissipated spherically in
waves, right? Isn't that what causes the resistance?

You were right, there's no direct analogy to the motion of an object
through air. That kind of aether theory has been disproven.

I was trying to point out that there is a direct analogy between an
object accelerating in an ultramundane aether and an object
accelerating in the air. The Lorentz transformation formula and the
Mach formula are identical and represent a condition where the faster
you go the more resistance there is to the motion to infinity (or the
speed of light and sound respectively). If you plot both formulas on a
graph, they have identical curves.

Does the math
really offer an explanation of why it takes less energy to move
tangentially rather than parallel? I think it has to be explained in
terms of a model. In a vacuum there would be no reason for the
resistance.


That depends on your model of a vacuum, doesn't it.

If you explain a vacuum as a kinetic distribution of space-time atoms,
then describe light waves in this medium, and then build a physical
coordinate system using these light waves, you should be able to
expalain the different energies required for the accelerations.

snip

What I find interesting is that Newton seemingly assumed a vacuum for
his laws of mechanics. That's understandable though because at the
time there wasn't any evidence that an ultramundane aether existed
and there wasn't any physical examples (like cyclotrons) to cause him
to think that there would be resistance to accelerating an object
(except from the mass of the object itself). But had Newton guessed
that we live in a medium environment even without the air, then his
laws of mechanics would have reflected resistance to motion because of
the medium. IMO, Maxwell and Lorentz both understood this.

Vern

Autymn D. C. wrote:
Ken, stop misspelling its, you retard.

shevek:
laboritory - laboratory


Thanks Autymn, I can't believe I misspelled it twice! AARGH!

wrote:
shevek wrote:
wrote:

snip

If a missile was propelled through an ultramundane aether which offered
resistance by a factor of gamma and a booster rocket was positioned on
the side of the missile, would the energy needed in the booster rocket
to change the direction of the missile be less than that in the main
rocket propelling the missile in the original direction?

Yes, that's what the equations above are telling us. However, if you
move to the frame of the missile, you will see that the energy needed
for acceleration (as measured in that frame) is the same in any
direction. The affect in question (more energy needed to accelerate
longitudal motion than transverse) is a coordinate-dependent effect.

Right, but forget the equations for a second and consider the
assumption of an ultramundane aether. In the frame where the aether is
isotropic, the missile is accelerating. Since more aether resistance
is encountered in the direction of the motion, more energy would be
needed to accelerate the object in this frame. If you were to
accelerate the object normal to the direction of motion, from the
aether frame, you would only be creating a small aether resistance as
the velocity w.r.t the aether in the normal direction is not nearly as
much as the velocity w.r.t to the aether in the original direction. In
the frame of missile, you would still find that forces were acting on
the missile, a greater force coming from one direction (the direction
in which in the aether frame the missile is moving) and a smaller force
from normal to that direction (in which in the aether frame you were
attempting to accelerate normal to the original direction). So, with
the assumption of an aether, the effect is not frame dependent, it just
changes from resistance against a moving object to an unknown force (or
an aether wind) acting on a stationary object.


Hi Vern,

That is one possible class of aether theories, the class where massive
particles are distinct from the aether, and are acted upon by the
aether. It sounds like you are envisioning here aether as only
existing -between- material bodies, rather than envisioning material
bodies as stable disturbances of the aether itself. As you may have
realized, I prefer the latter approach.

Incidentally, if the missile was at rest in the ultramundane aether,
and the laboratory was flying by, the observed effect would be the
same. The rest frame of this unltramundane aether is non needed in the
equation - hence this ultramundane aether is also superfluous as
pointed out by Einstein.

I'm suggesting a modification of Newton's laws to add in the effect
of aether resistance and I'm suggesting that without it there is no
explanation for the fact that it takes more force to accelerate an
object the faster you go and to also explain why it doesn't take as
much force to accelerate the object normal to its motion.


That fact becomes untrue when you measure the forces and accelerations
from within the rocket.

I see no need of -action- of the aether on material bodies in analogy
to air resistance. Rather, the effect is entirely due to the relative
motion of physical clocks and meter sticks through the aether. An
electromagnetic (physical) coordinate system ticks and extends at rates
proportional to the pressure tensor of the aether - motion through the
ether changes the pressure tensor, as described by Mach's formula of
gas pressure or the Lorentz transformation (as you pointed out below).
So in a way, it is analagous to air resistance.. but this aether
resistance manifests itself by changing the tick/extension rates of
coordinate systems in relative motion - not by providing an external
force on moving objects.

It's important to realize that these relativistic particles are not
acted upon by a force that slows them down - as happens to a projectile
in air. Relativistic particles in accelerators and cosmic rays do obey
Newton's law that an object in motion remains in motion unless acted
upon by an outside force.

I would say
yes because there is less aether resistance normal to the original
direction of the missile. However, it is a bit more complicated as the
motion of the missile through the aether can have a direct analogy to
the motion of an object through the air.

True again. The effect is not a result of "resistance" in analogy with
air restistance. There is no such thing, just as there is no air
resistance for a phonon moving across the room. Rather, the effect is
due to the changes in our coordinate system, which is built on light
propagation and the principles of relativity.

In this thread we're not talking about a wave, photon or phonon's
resistance due to motion w.r.t a medium, but instead we are considering
an object's resistance to acceleration. With the assumption of an
ultramundane aether, the effect is not due to the changes in the
coordinate system.


Let's pick an object as an example. How about an electron? In fact,
interference patterns can be observed with electrons - and they have an
associated deBroglie wavelength. Waves, wavefunctions, and solitons
are often good models of material objects.

Anyway, the effect must have something to do with changes of the
coordinate system, because some changes of coordinate system eliminate
the effect under discussion.

I should point out that the subject of what causes resistance to
acceleration (inertia / mass) in the first place, even without
relativistic effects, is also fairly wide open.


In such cases the excess
pressure from the object moving forward is dissipated spherically in
waves, right? Isn't that what causes the resistance?

You were right, there's no direct analogy to the motion of an object
through air. That kind of aether theory has been disproven.

I was trying to point out that there is a direct analogy between an
object accelerating in an ultramundane aether and an object
accelerating in the air. The Lorentz transformation formula and the
Mach formula are identical and represent a condition where the faster
you go the more resistance there is to the motion to infinity (or the
speed of light and sound respectively). If you plot both formulas on a
graph, they have identical curves.


I believe you are entirely correct that the similarity of these curves
is not a coincidence. They arise from the similar calculations,
determining the kinetic pressure in a moving reference frame. However,
a gradient in pressure in air will cause a force on e.g. an airplane
(air resistance). A gradient in 'pressure' of the aether causes no
such force on a neutral particle. It does affect charges and
electromagnetic forces in e.g. light, and therefore affects our
coordinate systems.


snip

What I find interesting is that Newton seemingly assumed a vacuum for
his laws of mechanics. That's understandable though because at the
time there wasn't any evidence that an ultramundane aether existed
and there wasn't any physical examples (like cyclotrons) to cause him
to think that there would be resistance to accelerating an object
(except from the mass of the object itself). But had Newton guessed
that we live in a medium environment even without the air, then his
laws of mechanics would have reflected resistance to motion because of
the medium. IMO, Maxwell and Lorentz both understood this.


We must be careful not to confuse resistance to acceleration with
resistance to motion. Aristotelan mechanics, and air resistance, are
examples of resistance to motion. Inertia and Newtoninan mechanics
suggest resistance to acceleration; SR suggests anisotropic (direction
& velocity dependent) resistance to acceleration, but no resistance to
motion.

Cheers - shevek

Autymn D. C. wrote:
shevek wrote:
yes because there is less aether resistance normal to the original
direction of the missile. However, it is a bit more complicated as the
motion of the missile through the aether can have a direct analogy to
the motion of an object through the air.

True again. The effect is not a result of "resistance" in analogy with
air restistance. There is no such thing, just as there is no air
resistance for a phonon moving across the room. Rather, the effect is
due to the changes in our coordinate system, which is built on light
propagation and the principles of relativity.

It's called viscoelasticity. With matter waves, they are perturbing
virtual particles when being moved.

Is this an explanation of inertia, or of relativistic mass originally
under discussion? In either case, a bit more would be helpful to help
us interpret your meaning.


In such cases the excess
pressure from the object moving forward is dissipated spherically in
waves, right? Isn't that what causes the resistance?

You were right, there's no direct analogy to the motion of an object
through air. That kind of aether theory has been disproven.

It's called terminal velocity. With matter waves, some of the
radiation is regenerated but they can only approach the driving speed.
The æther theory has not been disproven.


Well I apologize for saying that this kind of aether theory, or class
of aether theories, has been disproven. A more accurate statement
would be that experimental results are -extremely- difficult to
reconcile with that type of aether theory.

Again, I am talking about the kind of aether theories where the aether
exists only in the space between material bodies.

These theories don't make sense to me anyway, the whole point of aether
is to explain force fields - and force fields certainly exist inside,
and in some way are, material particles.

- shev