Relativity And Simple Algebra

[ralfcis]

I couldn't finish this, I will in a few days.

I must admit, after days of thinking I do not have a handle on this yet so I'm going to have to flail around a bit.

 

I think my last post was wrong, c is not a scalar value, it is indeed a velocity vector. Why do I suspect this? Well, the other two equations, that are based on Einstein's velocity combo law, yield the same results. Here they are again:

 

Y_ww = (v +u)(Y_vY_u) where Y_w = Y_vY_u(1 + vu/c) and w =(v+ u)/(1 + vu/c²)

 

DSR_w = DSR_v * DSR_u  where DSR_w² = (c-w)/(c+w) 

 

The sign of the velocities is positive if the direction is positive where the distance increases in the direction of the vector. The velocity w is the relativistic combination of v + u.

 

So Yc = Yv +cDSR where cDSR = v_ht which is the half speed velocity through time.

 

For example for v = .6c 

v_t = .8c (v_t is velocity through time = c/Y)

v_h = 1/3c (half of .6c)

v_ht = .5c (half of .8c) (also v_ht = c(c-v_h)/(c+v_h)) 

Edited December 22, 2019 by ralfcis

[ralfcis]

The math is leading to seemingly nonsensical results when plugging v=c into the above equations. 

 

First Y can't equal infinity, it must always be cancelled out in any useful equation. I proved this was true previously in this thread when I did the math for the twin paradox with the impossible scenarios of the twin returning at -c or accelerating away at +c in post #551. 

 

A brief summary of the result is that light travels at 2c with the velocity through space = c and the velocity through time also = c. Only for light is this true as a material object going near c would have its v_t approach  0. Time does not stop for light. There is a discontinuity at v=c where the results for material objects are very different for light at v=c. The math further showed that as Alice approaches c away from Bob after travelling 3 ly, she will permanently age 2 yrs more than Bob. If she returned to Bob at -c, she will permanently age 2 yrs less than Bob. Relativity has no comment on these issues which it deems are out of bounds.

 

Normally the velocity through time is an observed illusion controlled by the equation

 

c² = v² +v_t² 

 

where v_t = c/Y

 

So at v = .6c, v_t = .8c.

 

But during the time of relative velocity imbalance, which lasts from the time a change in relative velocity is made to the time it is received, the velocity through time is no longer an illusion and is no longer controlled by v_t=c/Y but by v_ht=cDSR  (v_ht is the half speed of v_t) . So what's happening is constant relative velocity has a constant Y but I don't know what form Y takes during an imbalanced relative velocity. Actually I do know what form it takes but I don't yet know the formula  relating cDSR in terms of Y. It's possible the formula is 

 

Y_v(c-v) = cDSR or Y_v(c+v) = c/DSR

 

Some other clues are that the lines of proper simultaneity have a rate of changing slope only during the velocity imbalance period. This rate of slope change may provide another significant formula. Another clue is the time lengths of the light signals multiplied by DSR yield Y.   

 

In the train rear example, the pink light signal from the middle of the platform to the rear of the train takes .625 yrs from t=0 where DSR = 2 in that direction and 1/2 for the yellow light signal direction which has a length of 2.5 yrs. So .625 * 2 = Y = 2.5 * 1/2. This mathematical pattern also appears in the formula

 

Y_ww = (v +u)(Y_vY_u) where Y_w = Y_vY_u(1 + vu/c²) and w =(v+ u)/(1 + vu/c²)

 

When you set u=c then w =c because that's the combo result of any v +c.

If w = c then Y_w = Y_u = Y_c 

So plugging in

 

Y_v(c+v) = c/DSR

 

into the above equation and cancelling out Y_w and Y_u you get

c=c/DSR which is no longer an equation.

 

So something went wrong and it's probable that Y_w, Y_u and Y_c are not equal when u=c. Just like in the example of Alice turning around at -c or changing velocity to +c away from Bob, Y_c is not infinity and Y_u has a finite value that is not equal to the finite value of Y_w. I don't know yet but my math senses say this is where to look.

 

The length of the light signals changes depending on direction (and hence the sign of DSR) and the depiction of the relative velocity. In the Minkowski depiction, light takes longer to reach something receding from it than it does something going towards it. The very definition of relative velocity says these are two different combined relative velocities yet the MMX found no difference. Just as DSR gives Alice the power to go many times c through time during the velocity imbalance period, so too could DSR give light the same power to adjust its velocity through time mathematically to accommodate velocities relative to it to keep c constant. 

 

The v_t for c is manifest in its frequency which is affected by the DSR. It's possible that when light is blue shifted or red shifted, these are the manifestations of c having its v_t adjusted by a velocity relative to it so that c remains constant. 

Edited December 22, 2019 by ralfcis

[ralfcis]

Ok this is a tough problem to solve and I love it. So far the problem with the equation is that infinity +1 is not equal to infinity and therefore can't cancel each other out as v approaches c. Let me show you what I mean in a numerical example.

 

v = 3280/3281 c. This is a velocity through space quite close to c.

v_h = 40/41c. This is the half speed velocity through space.

To check, plug v_h into the combo equation 

v = (40/41 + 40/41) / (1 + 40²/41²) = (2*40) / (41² +40²)/41) = (41* 2 * 40) / (41² +40²) = 3280/3281. It checks

vt = 81/3281c The observed velocity through time.

 

Plug these into the formula

 

v = v_h (1 + v_t/c) (or v_t = v_ht (1 + v/c) because v and v_t are two sides of the same coin and interchangeable but not how relativity defines this.)

 

3280/3281 = 40/41 ( 1 + 81/3281) = 40/41 (3281 + 81) / 3281

So 3280 = 40/41 (3281 + 81). It checks

 

But as v -> c we see a pattern in the equation

 

v -> oo/(oo + 1) where oo is infinity. Whatever number close to c you choose, v can always be written as v = n/(n+1). Yeah, mind blown relativists. I guess Einy missed this relationship between c and infinity. 

 

So since that is true for v, what is the formula for v_h using n.

 

v_h = n / ((n+1) + sqrt(2n+1)) 

v_h is always smaller number than vas v -> c, the "infinities" of the gammas for v and v_h have finite differences which can be exploited (the infinite parts can be cancelled out) to have valid equations when v=c. Now I need to show how this happens.

Edited December 22, 2019 by ralfcis

[ralfcis]

I find this unexpected but the formulas

 

v = n / (n+1) and v_h = n / ((n+1) + sqrt(2n+1))

 

work for any number (not just integers).

 

I was expecting it to work only for large integers. This means I'm not seeing a very fundamental underlying math and, like fluke-artist Einstein, have stumbled across something huge by accident without yet understanding it. 

Edited December 22, 2019 by ralfcis

[ralfcis]

Since v_ht = cDSR and v_t is interchangeable with v, let's re-write the formulas in a more useful form.

 

v_t = m / (m+1) and v_ht = m / ((m+1) + sqrt(2m+1))

 

Replace DSR with v_ht in the formula for velocity combination

 

DSR_w = DSR_v * DSR_u where w = c(1-DSR_w²) / (1+DSR_w²)

 

v_htw = v_htv * v_htu 

 

So if v = .6c, v_t = .8c and t_htv = .5c

and if v_u = .8c, v_t = .6c and t_htu = 1/3c

so v_htw = 1/6 c = c DSR_w 

 

Solving for combined velocity through space w we get 35/37c which is correct.

Let's check the results using m.

 

if w = 35/37, w_t = sqrt (c² - w²) = 12/37

 

w_t = 12/37 = m / (m +1)

so m=12/25 and (m+1) = 37/25

Plugging these into the top formula we get

 

v_htw = (12/25) / ((37/25) + sqrt (24/25 +1)) = (12/25) / ((37/25) + 35/25) = 1/6 and that is indeed the half speed of w_t = 12/37 which is the corresponding velocity through time for w = 35/37 with a DSR of 1/6.

 

Looks like an even simpler pattern is emerging:

 

If w =d/e and w_t = f/e, then v_htw = f / (e+d)

So for the example of v = 3280/3281, v_t = 81/3281 so v_htw = 81/ (3281 + 3280) = 1/81 which is correct.

 

Since we know what answers to expect, it should be easy to work backwards to derive the formal math.

Edited December 22, 2019 by ralfcis

[ralfcis]

So my math flailing has led me full circle back to these two equations:

 

v_ht = v_t / (1 + v/c) and its counterpart v_h = v/ (1 + v_t/c)

 

Since v_ht = c DSR and v_t = c/Y, the first equation can be re-written

 

Y*DSR = 1 / (1 + v/c) = F which I'll call the F factor.

 

As v-> c, F -> 1/2 when written as 1 / (1 + v/c)  but Y*DSR do not seem to approach 1/2. Y -> oo and DSR ->0 which is indeterminate.

 

How do you prove Y*DSR -> 1/2 and are not indeterminate?

 

Well Y = (c² + v_h²) / (c² - v_h²) and DSR = (c-v_h) / (c+v_h)

 

Y*DSR =( ((c² + v_h²) - 2cv_h ) / ((c-v_h)(c+v_h))) * ((c-v_h) / (c+v_h))

 

= 1 - 2cv_h / (c+v_h)² 

 

as v -> c so does v_h 

So

Y*DSR = 1 - 2cc / (2c)²  = 1/2 not indeterminate when v=c.

 

So I need to figure out how to use this result to scrub out the infinities in my velocity combo equations which are

 

Y_ww = (v +u)(Y_vY_u) where Y_w = Y_vY_u(1 + vu/c²) and w =(v+ u)/(1 + vu/c²)

or DSR_w = DSR_v * DSR_u where w = c(1-DSR_w²) / (1+DSR_w²)

Edited December 22, 2019 by ralfcis

[ralfcis]

Turns out to be a very neat little proof. We want to see as u->c what is the combined velocity w of v+u

 

c/DSR_v = Y_v (c+v) with u=c

Y_ww = (v +c)(Y_vY_u)

Y_ww =cY_u / DSR_v

 

From DSR_w = DSR_v * DSR_u we get

DSR_v = DSR_w / DSR_u

 

So w =  cY_u / Y_w DSR_v = c Y_u DSR_u / (Y_w DSR_w)

 

We know as any velocity approaches c, its Y DSR approaches 1/2 so the final result is

 

w = c.

 

It shows that cancelling out zeroes and infinities in an equation just won't work, they have to be finite real numbers. Specifically 1/oo does not equal zero as so many people think and the gamma function does not equal infinity when v=c. oo is not a real number, it cannot be mathematically manipulated as a value. It is like a math cancer that must be surgically removed from any equation.

Now we have to get back into what all this means for the MMX. (Michelson Morley experiment)

Edited August 16, 2019 by ralfcis

[ralfcis]

I know on physics forums, math is viewed with disdain and suspicion just like anything people don't understand. It's probably impossible to comprehend how a term Y that goes to infinity can be multiplied by a term DSR that goes to zero and the answer is 1/2. Those of you who know how to type in numbers on a calculator keyboard can verify for themselves the answer is correct. Let's use a velocity very close to c.

 

v=21523360 / 21523361 c = .99999995c

Y = 21523361 / 6561 = 3280.5

DSR = 1/6561

Y*DSR = 3280.5 / 6561 = .50000001

 

No matter how close you get to c, the result for Y*DSR (infinity times 0) is 1/2.

 

Now I'd think relativists with PHD's in physics would understand this. There are no infinities in physics that don't approach a finite number somehow. In this case as Y approaches infinity, DSR tempers that with its approach to 1/infinity and the result is their product approaches 1/2. 

 

PS.  v_ht = 1/2 v_t when v = c,  and v_h =c from the equation

v_h = v / (1+v_t/c) because v_t =0 when v = c.

 

Plug in v_h = 21523360 / 21529922 c into the velocity combo equation for the example above to check for yourself.

Edited December 25, 2019 by ralfcis

[ralfcis]

I just saw another crazy result

 

v_hx = v_x / (1+1/Y) since v_t = c/Y

 

 (F is not equal to 1 / (1+1/Y_v) ,F only applies to the equation v_ht = v_t / (1 + v/c) , not its counterpart v_h = v / (1 + v_t/c))

 

My previous conclusion that Y_c =1 has not been verified by the math above. The product of Y_cDSR_c is a finite value (= DSR_v =1/2) but what Y_c and DSR_c are individually has not been verified.

Edited December 25, 2019 by ralfcis

[ralfcis]

Sorry I've made an equation error in my last 3 posts which I will correct in red. I should have realized something was wrong when I concluded v_h = 1/2 v and v_h = v when v -> c. This contradiction can't be true under any circumstances and yet I tried to pass it off as true.

 

Turns out in post 917, if you change the value of v as u -> c, you don't get Y_u/Y_w -> 1/2, you get Y_u/Y_w -> DSR_v.

Edited December 25, 2019 by ralfcis

[ralfcis]

I have an opposite view to the science philosophers who can't do a lick of math that infest "physics" forums. Their perspective is that the math can be mistaken but the philosophy can't be so long as it agrees with observable and measurable phenomena. The foundation of the philosophy can also extend into explaining currently unmeasurable phenomena such as dark matter which is discussed at great length, for no productive reason, in "physics" (i.e. science philosophy) forums. 

 

The science history buffs may differ but I think Einstein was a turning point in science. Math was previously a tool to quantize and fit observation, Einstein showed observation can follow from the math. I think his philosophy is the wrong interpretation of the math so I've been primarily focused on math exploration and the math is pretty weird.

 

The first sign of weirdness is subtraction becomes division in that a negative velocity causes DSR to flip.

 

Another sign is the velocities through space and  time can be exchanged in equations.

 

What's even weirder is that velocities and gammas can be exchanged in equations. Here's how that works:

 

Certain terms keep appearing so I am going to assign them letters:

 

A = (v+u)

B= (1+vu/c²)

sqrt(c²B²-A²) = c/YvYu =v_tu_t

 

The last term shows a conversion between velocities and their gammas and a mathematical relationship between a square root subtraction of squares and an inverted multiplication. These terms and the new math where complex operators such as sqrts, squares, multiplication and division can all be expressed as addition and subtraction, can make complex relationships quite simple.

 

The complex relationships I need to reveal are:

 

1. an equation for Y in terms of constituent gammas only

2. an equation for w in terms of constituent gammas only

3. The complimentary expressions using time and space subscripts interchanged.

 

I think I have those answers but need to do extensive checking and derivations. Once I understand the interrelationships of the operators of this new math , I will have a tool on how to explain the emergence and significance of the MMX.

Edited August 31, 2019 by ralfcis

[ralfcis]

PS. If anyone is interested in shedding their math illiteracy, I will answer mathematical questions mathematically, But if you insist on philosophy or history discussions, I can't help you.

[ralfcis]

The math turns out to be very cool. It loops back on itself like a sphere that is completely self-enclosed and no matter what direction you take, you always end up at the same spot. It's like being on a planet at the edge of the universe yet in any direction you look, you're still at the 3D center of the universe (just like you're always at the 2D center of the surface of the earth. So here goes:

 

c² = v² + v_t²   The main equation that defines space and time

 

So  v_t = sqrt (c² -v² )

 

Y_vt = 1 / sqrt(1- (c² -v² ) / c² ) = c/v

 

so v = c/Y_vt  just like v_t = c/Y_v as expected because time and space terms are interchangeable in the equations.

 

We plug this result into relativity's velocity combo equation 

 

w = (v+u) / (1 + vu/c²) = c (Y_vt + Y_ut) / (1+ Y_vtY_ut)

 

It looks like gammas and velocities are also interchangeable in the equations but it doesn't mean you can swap t's and x's however you like.

 

Let's work out what w_t looks like in terms of u_t and v_t .

 

This equation previously derived:

 

Y_w = Y_vY_u(1+vu/c²)

 

w_t = c / Y_w = (v_t u_t) / c² (1+vu/c²) = (v_t u_t) / (1+1/(Y_vtY_ut))

 Notice the addition of v and u for w becomes a multiplication of v_t and u_t for w_t.

 

I think that's a complete enough picture for the conversion operations. Now I need to see what happens when u is set to c. Can I get around the infinity of Y_c to see what happens to the velocity addition of c and v and show another explanation for the MMX result.

Edited December 25, 2019 by ralfcis

[ralfcis]

So let's multiply these two equations together as good things always happen when you express velocities in terms of Yv.

 

w = (v+u) / (1 + vu/c²)

Y_w = Y_vY_u(1+vu/c²)

 

so Y_ww= Y_vY_u(v+u)

 

When v= u= 3/5,  w=15/17, Y_w= 17/8, Y_vY_u = 25/16, (v+u) = 6/5.

 

Let's boost u towards c by setting it to u= 4/5 and keeping  v= 3/5 for now.

 

so  w=35/37, Y_w= 37/12, Y_vY_u = 25/12, (v+u) = 7/5.

 

Let's double u to 40/41c 

 

so  w=323/325, Y_w= 325/36, Y_vY_u = 205/36, (v+u) = 323/205.

 

Let's re-write Y_ww= Y_vY_u(v+u)

 

to w/ (v+u) =  Y_vY_u / Y_w

 

As u -> c , the equation becomes

 

c/(3/5c+c) = 5/4 Y_u / Y_w

 So Y_u/ Y_w -> 1/2 (again!) 

Turns out in post 917, if you change the value of v as u -> c, you don't get Y_u/Y_w -> 1/2, you get Y_u/Y_w -> DSR_v.

 

Let's check the numbers for u = 40/41c

 

(323/325 / (3/5 + 40/41)) 4/5 = .5046

 

So infinity over infinity is limited to Y_u/Y_w =DSR_v and the gamma of the combined velocities when one of them is c is Y_w =Y_u/DSR_v  that of the velocity going to c.

 

Even though both w and u velocities are technically c this result shows there's a difference between c's. c can morph into relative multiples of c and still remain c from each perspective. 

 

We've already seen an example of going many times c (through time) in the twin paradox but now we're going to see how this also happens in the MMX. Sad no one else can share in the beauty of how this multilayered lotus flower is opening up.

Edited December 25, 2019 by ralfcis

[ralfcis]

Important

 

Another fn mistake. If you change the value of v as u -> c, you don't get Y_u/Y_w -> 1/2, you get Y_u/Y_w -> DSR_v. Infinity over infinity does not tend to a fixed fraction but one that is highly dependent on v. That makes way more sense and shows how deeply interrelated the math is.

 

So I need to bring in the formula:

 

c/DSR_v = Y_v(c+v) and work it into the equation:

 

_c / (v+c) =  Y_vY_u / Y_w  =  c/(c/Y_v DSR_v )

 

So  DSR_v = Y_u / Y_w    Yes! Correct!

 

This formula is the holy grail. It explains why the length of light signals change due to their direction. In the train station example, the light lines towards a train going at .6c towards the station were only half as long as the light lines going from train to station within the train. The light signal inside the train corresponds to the light signal inside the relatively stationary tube for the MMX and the light signal from the platform to the train corresponds to the moving tube relative to the aether in the MMX.

 

Since Y = c/v_t the light signal in the moving tube at .6c has double the velocity through time as the stationary tube which erases the time effects of the tube's velocity on the total relative velocity of the light signal towards the end of the moving tube. I'm just spitballing, I still need to mathematically verify what i just said.

 

Too bad all the forumorons got off the bus but, in truth, the trip wasn't going to change them anyway. Oh, and I may be premature but Einstein's explanation for the MMX really looks kind of clumsy now. That's not going to help me but I really enjoy sticking it to relativists.

Edited December 25, 2019 by ralfcis

[ralfcis]

I had decided that I was going to spend all day today explaining the physical relevance of the math I just presented but it's 4:46 PM and I haven't even started yet. I hope I can get to it sometime tonight because it's some really exciting stuff.

[ralfcis]

Ok today is the day. I just hope I'm still not sitting here 10 hours from now. There will be some math but only at the beginning.

 

Relativity, like any math based physics, has only 1 goal: to balance the equation. It generates some very counter-intuitive physical phenomena to ensure this. The number one rule is that nothing can go faster than c (whether through space, time or spacetime needs clarification) and it's enshrined in two forms of the same equation. Other forms of the 1 equation generate 4 main terms (v, v_t, v_h, v_ht) whose mathematical  interactions generate all of relativity's phenomena. Einstein's theory had a completely different take on the math to describe (and misinterpret) the same phenomena.

 

The main equation is

 

 c² = v² +v_t²  also written as the gamma function Y = c / sqrt(c²-v²) ( v_t = c/Y and v = c/Y_t)

 

The sqrt of the sum of the squares of velocity through space and time can't exceed c from any perspective. The corollary to this is that everything moves at the composite velocity of c from any perspective. Everything moves at the same composite speed through spacetime where the velocity through time regulates this and shrinks as the relative velocity through space increases. Since light's velocity through space is at the max, then c through space is the same from all perspectives as is c through spacetime from all perspectives.

 

So let's modify the rule to state the max speed through space is c but everything except light moves at the combined velocity c through time and space and both versions of c are invariant from all perspectives. As I will show, this is more a suggestion rather than a rule because it is broken in at least two cases. Light can move through time at multiple c as needed which seems to break the primary equation. When matter changes relative velocity, it can also go at multiple c through time. But let's leave these exceptions to the side for now.

 

Relativity is so concerned about maintaining the speed limit (so that causality remains invariant) it comes up with a second form of this equation such that the addition of two velocities through space can't exceed c through space. A corollary to this is the addition of any velocity through space with c results in c. Since all velocities are composite velocities through space and time, we should be able to prove both forms of the equation are actually the same equation. Here's the proof:

 

c² = w² + w_t²  where w is the combined velocities of v and u which are also composite velocites: c² = v² + v_t² , c² = u² + u_t² 

 

Instead of deriving w and w_t from the main equation, we already know the formulae and we can reverse derive them from the math I'm presenting here.

 

w_t = c / Y_w = c / (Y_vY_u(1 + vu/c²)) and w =(v+ u) / (1 + vu/c²)

 

So jumping ahead and sticking this into the main equation we get

 

(c² +vu)² = c²(v+u_x² + c²/Y_v²Y_u²   skipping steps

 

c⁴ +v²u² = c²(v²+u²) + c²/Y_v²Y_u²  skipping steps

 

c² = (v²+u²) + ((c²-v²)(c²-u²) - v²u² ) / c²  skipping steps

 

c² = (v²+u²) + c² - v²-u² 

 

c² =  c²  So yes c is limited to c whether you add two velocities through space, two composite velocities through spacetime or a velocity through space and it's velocity through time from any perspective.

 

Instead of just accepting c is constant from all perspectives because there is only 1 composite relative velocity which is c, Einstein felt the need to mess things up with his needless concept of time dilation and length contraction working together to keep c constant. This led to nonsensical results one of which is a photon covers the universe instantaneously from birth to death. It also led to a fudged explanation of the MMX as I showed in my train example.

 

More to come.

Edited December 25, 2019 by ralfcis