Relativity - Affine Geometry ?
Einstein's special theory of relativity is about measurements between frames of reference in a state of non acceleration. It is an observation that the velocity of light is independent of the relative velocity between observer and the source. From this Lorenz postulated a length contraction to allow for this. Einstein derived the same equation from the observations.
If you look at this equation there is no special frame so any frame may be taken as the reference frame.
Let us consider the passage of mesons from the origin in the upper atmosphere to their arrival at a counter on the earth's surface. The observer "sees" the origin from his frame and since he is "looking" at the meson then this point is only a few feet above the laboratory roof because of the Lorenz contraction. Thus accounting for the short passage time. The "meson" "sees" the path it travels from the upper atmosphere to the surface as the same few feet as it is "looking" at the observer's reference frame.
The Lorenz transformation
It is just a simultaneous equation.
The invariance of the velocity of light is an experimental result. It means that when you measure the velocity of light some how the result is independent of the velocity of the source, like a star or a lamp on a moving bench.
If you can work out the time an aeroplane takes for a return journey west to east with a constant east wind then you are well on the way to calculating the Lorenz transform.
Suppose the aircraft does v knots through the air and the wind is u knots over land then going from west to east d nautical miles the velocity is v-u and it takes d/(v-u) hours to get there going back it takes d/(v+u) hours. so the average is (d(1/v+u)+d(1/v-u))/2 =
With the Lorenz transform c is constant so if the velocity of the source is v then to make the velocity c in both frames the distance the light travels must be contracted to make the time d/v once again. So the rest distance is d, if d* is the contracted distance then d*/c=d/c+v and the same for the source with a velocity -v so d*/c=d/c-v so (d*/c)^2=d^2/(c+v)(c-v) so,
This contraction is not real it is an effect like perspective because we use light to measure the distance, it has no physical meaning.
Quite simple really, it is an example of a geometric mean and is called a metric as in differential geometry where you look at a hill, going over the hill is further than along the ground and if you are unaware of the slope the longer distance would be the only way you could know of the hump. Relativity is like that, we cannot see the real path of the light it is distorted by the space distortion caused by the movement of the source or observer. Except that in relativity the path over the hump is shorter that the path along the flat path. Think of a space time diagram the path is a ^ but the path over the ^ is shorter at high velocities than the path and very slow velocities.
A six inch bar is still a 6 inch bar there is no
contraction whatsoever. It is demonstrated each time you use an electric motor
or switch on an electric lamp. The electrons move the positive charges do not,
this means that the electronic length is shorter than the bar of metal they
are moving in but no positive charge occurs at the end so there is no
contraction at all.
There is no contraction really and no upper limit on velocity. If you work out the momentum it Mv/(sqr(1-(v/c)^2) but when you separate it for integration you go M (v/(sqr(1-(v/c)^2) the mass does not increase only the number v/(sqr(1-(v/c)^2) and this is the real velocity V.
Much more fun you can get velocities higher than light with a rocket or accelerators where the phase velocity of the travelling wave (as in a rhombadron) is higher than light. Remember group velocityXphase velocity is c for light in a waveguide and the same is true of matter waves so a stationary electrons phase wave occupies the whole universe and the energy quanta of a free electron is influenced by the size of the universe and all the matter in it.
The increments of energy that an electron is given as it accelerates is in reality determined by the size of the universe. This is because an electron is a group of waves of infinite velocity that extend around the 4-sphere universe. When the group changes its velocity the change is reflected by an alteration in the diffraction pattern as in a boxed particle and a quanta of radiation is emitted that has a frequency determined by the difference in frequency between the frequency of the initial step and the frequency of the final state.
This energy comes from the electric field that pushes it and since this push is by a virtual photon the emitted photon is the realised virtual photon of the electric field brought into existence by the movement of the electron.
When the universe was very small the steps were very large and so the universe was actually very cold at it conception as the quanta were too large for anything to move.
It seems to me that there is no real contraction, only an effect similar to perspective as a result of the way light travels and is only an appearance.
Relativity is 4-space perspective.
This would mean that a journey in a very fast space craft to a nearby solar system (say 10 light years) would take a time given after the length contraction and time contraction had been allowed for. (This is called the 4-velocity) I suggest that the 4-velocity is the "real" velocity and what we see is the result of 4-space perspective.
It would therefore be possible for deep space journeys with a fast spacecraft in normal time spans. (By fast I mean velocities close to that of light - say closer than 99%c).
It also strikes me that since all frames are equivalent the "Twin Paradox" does not occur, the two brothers will agree both about the time and distance travelled.
Since perspective is the projection of 3-space to 2-space then relativity is the projection of 4-space to 3-space.
I made a mistake when I changed the co-ordinates from linear to exponential to look inside a black hole. You see the scale is similar to the Kelvin scale. By using a linear scale we can see 0 but it cannot be reached. We should really be using a log scale so 0 is – infinity so cannot be reached.
The theory of the black hole is wrong in the same way.
To understand it you need the theory of metric spaces and the Einstein’s concept of sitting at the foot of a cliff. Now light is falling down towards you and it gains energy as it falls but since it cannot go any faster (or slower) it increases in frequency, so shortening the wavelength. This wavelength can be thought of as the shape factor of space at that point and the frequency as the local shape factor of time.
Going up the same thing applies.
So space is smaller as you go towards a massive object (like your left index finger). Now as a small object goes near a big object the space gets changed in length, (it is not real, it is 4-perspective), so like a beam of light passes through a change in refractive index it is bend towards the region of space is shorter. It is as if light moves more slowly.
if now is 15,000 km from earth's centre, and soon is 14,991 from the earths centre then a potential energy change takes place of Gm/15000-Gm/14991 G is the universal constant of gravity (measurable) and mass is the mass + energy within the point.
So the metric means that as we do the translation the space shape factor changes
Now the shape factor of space is = Gm/r as far as I understand it.
so far away the factor is 0
The light here is of frequency f and wavelength L=cf
at a distance r the shape factor is Gm/r so the energy of light will be higher. The energy of light is hf in flat space but as it goes closer to a mass + energy it is of higher energy, Gm/r the light gains potential energy (Gm/big to GM/less big)=hf to hf + small bit
so Gm/r2-Gm/r1=hf1-hf2 to put that into the differential form for integration is not needed if we use the metric tensor. the shape has changed from (Gm/r2)/h to (Gm/r1/h) to get the length we have to use the wavelength this is fL=c so L=c/f
So the length of space has changed from c/(Gm/r1/h) to c/(Gm/r2/h). So space can be mapped in shape according to the location relative to massive objects or areas of energy, includes light energy.
Relativistic momentum is m (v/(sqr(1-(v/c)^2). The mass is invariant.
The mass ratio for the photon rocket is = exp (1/c( integral (v/c)/sqr(1-(v/c)^2))dv) This comes to 3.56 for v=0.999c
I used this because it makes the sum dimensionally correct. It is to do with the integration.
V=v/(sqr(1-(v/c)^2)) DV/dv has a c in it.
Einstein addressed the twin paradox in special relativity in a relatively unknown, unusual and rarely cited paper written in 1918, in the form of a dialogue between a critic and a relativist. Contrary to most textbook versions of the resolution, Einstein admitted that the special relativistic time dilation was symmetric for the twins, and he had to invoke, asymmetrically, the general relativistic gravitational time dilation during the brief periods of acceleration to justify the asymmetrical aging. Notably, Einstein did not use any argument related to simultaneity or Doppler shift in his analysis. I discuss Einstein's resolution and several conceptual issues that arise. It is concluded that Einstein's resolution using gravitational time dilation suffers from logical and physical flaws, and gives incorrect answers in a general setting. The counter examples imply the need to reconsider many issues related to the comparison of transported clocks. The failure of the accepted views and resolutions is traced to the fact that the special relativity principle formulated originally for physics in empty space is not valid in the matter-filled universe. Einstein's special theory of relativity is about measurements between frames of reference in a state of non acceleration. It is an observation that the velocity of light is independent of the relative velocity between observer and the source. From this Lorenz postulated a length contraction to allow for this. Einstein derived the same equation from the observations. If you look at this equation there is no special frame so any frame may be taken as the reference frame. Let us consider the passage of mesons from the origin in the upper atmosphere to their arrival at a counter on the earth's surface. The observer "sees" the origin from his frame and since he is "looking" at the meson then this point is only a few feet above the laboratory roof because of the Lorenz contraction thus accounting for the short passage time. The "meson" "sees" the path it travels from the upper atmosphere to the surface as the same few feet as it is "looking" at the observer's reference frame.
End of Quotation
It seems to me that there is no real contraction, only an effect similar to perspective as a result of the way light travels and is only an appearance. Relativity is 4-space perspective. This would mean that a journey in a very fast space craft to a nearby solar system (say 10 light years) would take a time given after the length contraction and time contraction had been allowed for. (This is called the 4-velocity) I suggest that the 4-velocity is the "real" velocity and what we see is the result of 4-space perspective V=v/sqr(1-(v/c)^2) V is the 4-velocity and the velocity in all frames. It would therefore be possible for deep space journeys with a fast spacecraft in normal time spans. (By fast I mean velocities close to that of light - say closer than 99%c). It also strikes me that since all frames are equivalent the "Twin Paradox" does not occur the two brothers will agree both about the time and distance travelled.
Since perspective is the
projection of 3-space to 2-space then relativity is the projection of 4-space to
So even the man himself admitted his idea had been misinterpreted by his followers. The general theory relativity that deals with acceleration does not map well into special relativity because the forces on the objects do not influence signal exchanges in any way.
If you think of the star ship journey in terms of a two dimensional space time diagram (other massive objects are too far away to be of any influence) then you have a hill. Now in normal hills the distance over the hill is longer than the distance along the flat, but in the relativity case the distance over the hill is shorter than the flat because it is a negative hill.
That is because one of the dimensions is jct and the other is x now using Pythagoras the hypotenuse for this negative hill, which is really just a wobbly 'v' is 2*( x^2-(ct)^2)^0.5 and that is less than 2x. This is because we are in hyperspace. x is the "rest" distance and "t" is the time calculated at the x/v where v is the velocity. So the distance travelled is 2*(x^2-(c(x/v))^2)^0.5. and that takes (that distance)/v in real time.
The precise shape of the curve only means we need to do an integration along the curve VIS differential geometry, I get it now, with my brand new brain, fitted after they took out my old one a couple of months ago.
Get it! So space travel is really very easy you just have to fast enough.
"The faster you go, the quicker you get there" just like running a race!
I didn't win!
My understanding of special relativity is limited. However my opinion is that the contraction effects are a 4-perspective effect due to the constancy of the velocity of light, there is no physical contraction at all.
My mind forms a concept of a ^ shaped hill of the space-time trajectory of a return particle path where the path over the hill is shorter that the path taken at rest or very slowly, so the higher the velocity the shorter the path over the hill. It is a metric space.
The other related point is the relativistic momentum now since the path depends on the velocity then there is the notion of a real or true velocity v/gamma. In the momentum of a particle the momentum as measured by collision is m v/gamma so as the velocity increases the mass remains the same.
In the derivation of the kinetic energy the formula:
may be separated into to partial fractions
m x v/Sqr(1-(v/c)^2)
then you integrate with respect to v from 0 to a number less than c to find the kinetic energy.
This gives the solution mc2 for the rest mass energy as with the older Einstein version. He chose his method because of theological augment to make the maths easier: he said "God would not choose such a ‘complicated’ mathematics". God is not stupid either.
This method gives the idea of a true or real velocity of v/sqr(1-(v/c)^2). However the time taken for a round trip works out different from the usual relativity theory, as in a Metric space.
The general theory is a metric space where the distance shortest between adjacent points is given by a metric resulting from the sum mass + energy nearby. It could be represented by a matrix which could include the special theory as terms in the matrix to work out the distance between adjacent points.
Again there is no real contraction it caused by 4-perpective due the metric caused by the constancy of the velocity of light.
The metric is the way (a formula - a set of rules) we calculate the distance between points in a manifold. In ordinary Euclidian 3-space it is given by Pythagoras: s^2=x1^2+x2^2+x3^2. In Einstienian 4 space it is s^2=x1^2+x2^2+x3^2-(ct)^2 That is when nothing in the description is moving. When things move the usage is to measure distances, usually of the observers frame.
The metric tensor then contains the elements that are involved is special relativity, transforming measurements from one frame to another. It transforms one 4-vector to another 4-vector in the general case which includes acceleration and gravity then the metric tensor contains these elements as well.
But it still transforms from one 4-vector to another 4-vector in matrix multiplication.
V2=V1 x M where M is the metric tensor.
Einstein's idea was about light coming down from a lamp on a cliff as the light came down it gains energy from the gravitational pull so that its energy increases, Light cannot change velocity so the frequency rises to h(f+df) so that h(df) is the increase, this energy comes from the fall sp hdf=Gm/r1-Gm/r2 where r1 is the position of the light and r2 is his eye. Now since this increase changes the colour of the light it is a blue shift. When a light is shone up to the ledge a red shift occurs.
Since f=c/L where L is the wavelength the shape of space is smaller at the bottom of the cliff than on the ledge.
It is the same as a clock because frequency of light is like the ticking of a clock.
h(c/L)=Gm/r where r is the bottom of the fall from infinity. So the length of space 1/L=Gm/hcr:
L=hcr/Gm hc/G is constant m is the central mass and r is the distance from the centre of the spherical mass.
For 3 D a matrix must be used because the effect depends of the direction of the vector it operates on.
The combination of the special and gravity makes the general theory and a metric tensor (a 4 by 4 matrix) with 4 vector being the V1 and M the metric tensor and V2 the exit vector. The values in the metric are formulas based on the position of the mass and the velocity of the projectile, powered trajectories require to include dynamic energy factors.
In the differential form the transformation is between adjacent points in 4-space and using Fermat's theorem the path of least time can be found and this is a geodesic in the non Euclidian space near a massive object of a projectile moving at relativistic velocity - close to c.
Another important path is where the projectile is powered by a rocket and here the object is accelerating.
I would have thought total energy description - langranian type description may be easier in conjunction with Fermat's theorem of least time to compute the trajectory in space time as a series of 4-vectors.
Space means “nothing”, how can nothing curve, how can there be a hole in nothing and how can nothing flow into a hole in nothing?
I was given this philosophy by a Congregational Priest. It hit me like a brick through the head…
Gravity is different from that, I think.
Special relativity is about objects (not space) moving at constant velocity and the notion of relativity is that there is no reference point and that velocity can only be measured relative to another object. That it is why it is “relative”.
The other stem is the observation that the velocity of light (in other words the thing we use for measurement) is the same when measured from any object that can have any velocity relative to any other object. The special theory considers measurements using light taken on one object (in no way special) about another object that is in constant relative motion to the first. And the other way round.
The general theory includes acceleration. Einstein’s great leap was that Gravity is equivalent to acceleration.
However the same theory results if you guess that gravitational influence moves with the velocity of light and in other ways similar to light, like its velocity of influence is constant irrespective of the relative velocity that the gravitating objects under discussion.
This indicates that gravity is in reality electromagnetic in nature.
There is no space.