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## onsdag 28 augusti 2019

### The True Paradox: The Twins Have the Same Age at Reunion

The twin paradox has been haunting physics for 108 years without any resolution in sight.

Let us approach the paradox using the basic physics of a clock in the form of a harmonic oscillator as the most basic model of all of physics. Consider two twins A and B at rest in a Euclidean $(x,y,z)$-system both equipped with identical clocks in the form of harmonic oscillators acting along the $y$-axis initiated in exactly the same way.

Let B take off into a journey along the $x$-axis to a distant point and back again.

Let A and B compare the readings of their clocks when B returns home. What will they find?

1. Einstein's (special/general) theory of relativity predicts that B will be younger that A. Since ageing is measured by clocks, this means that B's clock lags behind A's clock at reunion. B's clock may show 1 year while A's clock shows 2 years, if B's has aged 1 year and A 2 years during B's roundtrip. That's Einstein's prediction!

2. According to basic physics the performance of a harmonic oscillator acting along the $y$-axis cannot be affected by motion along the $x$-axis, not by uniform translation nor by acceleration/retardation. This means that B's clock during B's roundtrip will work exactly the same way as A's clock and therefore will show exactly the same time when B comes home. This means that A and B will have exactly the same age at reunion.

So, basic physics of clocks as harmonic oscillators shows, without any possible doubt, that A and B will have the same age, while Einsteins theory of relativity predicts that B will be younger.

This is a true contradiction or paradox, which shows that Einstein's theory of relativity cannot be correct. A correct theory about physics cannot give a prediction which contradicts the completely basic uncontroversial physics of a harmonic oscillator.

The twin paradox of unequal ageing thus is only an illusion (of Lorentz "time dilation").  This "resolves" the twin paradox, but the consequence is that Einstein's theory of relativity is also only an illusion.

It is a true paradox that this illusion has come to form the basis of modern physics.

PS1 When I ask a group of physicists to comment on my twin paradox posts, I do not get the reaction that what I am saying is wrong, but instead an excuse that they are too busy with other aspects of physics to comment, while admitting that the question is of fundamental importance to physics of today, and instead kick the ball back by saying that if I myself try a little harder, I will find a resolution.

PS2 The marine chronometer constructed by John Harrison in the mid 18th century working at a steady rate independent of travel over sea allowed for the first time precise determination of longitude. The key was to keep away from time dilation.

## måndag 26 augusti 2019

### Einstein's Biggest Blunder: Mixing of Space into Time

Einstein "biggest scientific blunder" in his own view was the introduction of a zero-order term with coefficient $\Lambda$ named the cosmological constant in his cosmological field equations as a fix to get a stationary universe.

But an even bigger mistake/blunder was to change the view of the mixing of space into time expressed in the Lorentz transformation from that of Lorentz as a formality without true physical meaning, into a reality of space-time with space and time on equal footing in his 1905 article presenting his special theory of relativity SR to the world. Einstein took this step in sharp contradiction with the view of Leibniz with
• space = order of coexistence
• time = order of succession.
Einstein had a poor understanding of mathematics, which apparently allowed him to believe that just because (1d) space can be ordered along a spatial coordinate axis with space coordinate $x$ as a real number, and time can be ordered along a temporal coordinate axis with time coordinate $t$ as a real number, and the coordinate axes of real numbers for $x$ and $t$ superficially look the same, (1d) space cannot be distinguished from time. Einstein expressed this revelation as
• Time and space are modes by which we think and not conditions in which we live.
• Space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind union of the two will preserve an independent reality.
Einstein thereby left the physics of Leibniz with a sharp distinction between space and time, into a world of imagination without distinction and thus physics. This was Einstein's biggest mistake!

It led Einstein to work with "events" without spatial extension for which the essential aspect of coexistence had no meaning and physics was lost. In particular, it led Einstein to a special theory of relativity without physics based on a confused derivation of the Lorentz transformation incorrectly assuming that two different light pulses with spatial extension emitted in two different systems are one and the same light pulse without spatial extension.

What is so completely amazing is that Einstein's mistake of mixing space and time has become a religion for modern physicists. The twin paradox shows that this is confusing fake physics. No wonder  that modern physics is in state of deep crisis, caused by Einstein in particular.

## lördag 24 augusti 2019

### Fake Resolution of Twin Paradox

The twin paradox of special relativity SR has traumatised physics ever since it was first formulated 100 years ago, and several attempts to resolve the paradox have been presented over time, none of which has been acclaimed as the correct resolution. Let me here show that a popular variant, which many physicists cling to, is a fake resolution.

We recall that the special theory relativity connects the space-time coordinates $(x,t)$ and $(x^\prime ,t^\prime )$ in two inertial systems moving with constant velocity $v$ with respect to each other by the Lorentz transformation:
• $x^\prime =\gamma (x - vt)$, $t^\prime =\gamma (t - vx)$,
• $x =\gamma (x^\prime + vt^\prime )$, $t =\gamma (t^\prime + vx^\prime )$,
where $\gamma = \frac{1}{\sqrt{1-v^2}}$ assuming the speed of light is 1 and $\vert v\vert \lt 1$. We see that the (1d) space coordinate $x$ and time coordinate $t$ appear in symmetric form with an apparent similarity between space and time, which Lorentz viewed to be a formality without physics, but Einstein took as a basis of modern physics with space mixed into time.

Consider now two twins, twin A fixed at the origin $x=0$ of the $(x,t)$-system, and twin B fixed at the origin $x^\prime =0$ of the $(x^\prime ,t^\prime )$-system.  A's clock reads $t$ and B's clock reads $t^\prime$. A will see B follow the trajectory
• $x=vt$,
and so at time $t=1$ say, A will see B at $(v,1)$ in the $(x,t)$-system, while the corresponding coordinate in the $(x^\prime ,t^\prime )$-system is $(0,\gamma (1-v^2))=(0,\frac{1}{\gamma})$. A's clock thus reads $t=1$, while B's clock reads $t^\prime =\frac{1}{\gamma}\lt 1$. Twin A (stationary) thus finds B's clock (moving) to run slow compared to A's clock with the factor $\frac{1}{\gamma}$.

On the other hand, trajectories in the $(x,t)$ system of constant $t^\prime$ take the form
• $t=vx + constant$
with the line $t=vx+(1-v^2)$ passing through $(v,1)$. Setting here $x=0$ we find
$t=(1-v^2)=\frac{1}{\gamma^2}$ to be the reading of A's clock on the trajectory of constant $t^\prime$ through the switch point $(v,1)$. B thus views A's clock to read $\frac{1}{\gamma^2}$ when B's clock reads $\frac{1}{\gamma}$, and so B views A's clock to run slow by the factor $\frac{1}{\gamma}$

So far, A considers B's clock to run slow, and B considers A's clock to run slow by the same factor, but since A and B will never meet the contradiction can be viewed to be only apparent and thus not really paradoxical.

To make A and B meet a change of mutual velocity must be made. Assume then that B switches direction at $t=1$, to rejoin A at $(0,2)$ in the (x,t)-system. This means that B changes/jumps to a new inertialsystem $(\bar x,\bar t)$ related to the $(x,t)$-system by the above Lorentz-transformation with $v$ replaced by $-v$. By symmetry, the trajectory of constant $\bar t$ passing through the switch point $(v,1)$ will cross $x=0$ at time $t=2-(1-v^2)=1+v^2$.

B thus views A's clock to run slow both before and after switch, but can come to a view in agreement with the reading $t=2$ of A's clock at reunion by assuming that A's clock takes a jump of $2v^2$ at the switch.

Note that A and B must agree on the readings of both their clocks at reunion. It is not enough that only A views B to be younger. The only way B can accept this is to assume that A's clock takes a jump at the switch to a new inertial system.

Summary: A considers B's clock to run slow by the factor $\frac{1}{\gamma}$, and vice versa.
Despite the fact that B sees A's clock running slow, B can come to agree with A at reunion that A is older, by assuming that A's clock takes a jump when B switches from one inertial system to another.

• How it is possible that B's clock at reunion can be seen to run slow compared to A's clock, when B thru the whole round-trip sees A's clock running slow?
The proposed resolution:
• This is possible if B assumes A's clock to take a jump forward when switching from one inertial system to another.
Is this a good resolution within SR? For B to assume that A's clock is reset with a sudden jump forward to give the impression at reunion that A is older? Can you change age by resetting a clock?

Of course not! The resolution is a fake resolution going outside SR by switching inertial systems and thereby resetting A's clock as viewed by B. It is amazing that such an obviously false argument can be put forward by physicists.

What then about going outside SR to general relativity GR, bringing in the idea that at switch B undergoes both retardation and acceleration with unknown effects, such as making A's clock jump forward? This can only make the argument more false and unphysical.

What do you think? Is the proposed resolution true physics or fake physics? Are you a true physicist or a fake physicist?

I have asked a group of physicists to comment the post and its truth value.

Bottom line 1: If you identify your ageing with the reading of your clock, you can by resetting the clock take on any age. If you think this is real, you have a problem.

Bottom line 2 Am I speaking about a petitesse, which physicists can dismiss as crackpot petitesse which can only be met by silence? No, the twin paradox is a real paradox of SR, which if unresolvable will destroy SR as physics. Therefore physicists must come up with a resolution or face the consequence, and that is far-reaching.

Connection to Dingle: Recall that Dingle posed the question to leaders of the physics community how it can be that both twins age more slowly than the other, a tough question which was side-stepped into the above fake physics argument with B switching inertial system and resetting A's clock. Dingle posed his question 50 years after the twin paradox was formulated and now another 50 years has passed with the paradox as glaringly unresolved as ever with silence as the only reaction from leading physicists. No wonder that that modern physics is in a state of deep crisis.

PS The Twins Clock Paradox History and Perspectives by by RL Shuler Jr (2014) identifies more than 200 articles over a period of 108 years seeking to resolve the paradox using at least 10 different approaches, however without ever reaching a conclusive answer:
• ...there is no doubt at any rate that the twins or clock paradox continues to fascinate and confuse the public and physics students . Much of the literature seeks more effective ways of explaining or visualizing Special Relativity (SR). Though SR does not actually convey longer life experience, the ability to affect clocks, presumably in some explanations remote clocks, seems magical. What causes this effect? Is it the acceleration or the travel at high velocity? Even after 108 years, it can seem to depend on which paper one reads.
This is also my experience from posing the question to physicists and receiving no meaningful answer. Since apparently after 108 years the paradox has not been conclusively resolved, the chance to do so over the next 1000 years appears slim...but of course physicists claim that steady progress towards as resolution is being made, and that therefore the paradox is no real threat to SR, only a nuisance from crackpots seeking to create trouble and mistrust by spreading disinformation.

If there are 10 different suggested resolutions and no agreement, what is the chances that all are correct, or only one? Not big!

Of special concern to Shuler is the difficulty to explain the resolution of the paradox to students witnessed by many physics teachers. It seems that no matter what explanation is tried, students  complain that they cannot follow the argument and say that they understand. Is the reason that students are stupid or that the argument is fake physics? You can test yourself: Is there any of the 10 different resolutions of the paradox, which you can follow and say that you understand to be correct? If you say no, does that mean that you are too stupid to follow even a simple basic argument about physics?

## måndag 19 augusti 2019

### Basic Questions Without Answers Corrupts Science

If you have a curious mind, you may have asked the following questions to a professional aerodynamicist/physicist:
1. What is a correct explanation of the flight of an airplane or bird?
2. What is a correct explanation of the twin paradox of the special theory of relativity?
If not, try it yourself! I have done so and met the following first response:
• This was settled long and can be found in many books.
When you ask for details, you get (if your are lucky) pointed to a variety of mutually contradictory explanations, each one viewed to be incorrect by some group of scientists, but no explanation which is generally agreed to be correct. Many incorrect explanations, but no correct.

When you point this out, you are met first with frustration, then with silence. This is unfortunate because unanswered basic questions, which are swept under the carpet, fosters corrupt science.

If you are curious you find answers to the questions on this blog.

## lördag 17 augusti 2019

### Galilean Relativity as Many-Minds Relativity

Many-Minds Relativity MMR explores an alternative to Einstein's special theory of relativity SR. The basic setting in both MMR and SR concerns observations made in different spatial Euclidean coordinate systems moving with constant velocity with respect to each other.

In MMR observers share a common time $t$ (Universal Common Time UTC as in the GPS system), while space coordinates in a system $X^\prime$ with space coordinates $x^\prime$ moving with velocity $v$ respect to a system $X$ with coordinates $x$, are connected by the Galilean transformation
• $x^\prime = x- tv$
stating that the origin of $X^\prime$ moves in $X$ with velocity $v$.

In SR there is no common time and space-time coordinates are connected by the Lorentz transformation mixing space into time in violation of Newtonian mechanics. SR leads to many strange effects such as "time dilation" and "space contraction" as effects of motion with constant velocity, which are deeply paradoxical.

1. All observers share a common time set by synchronised cesium clocks according to the SI standard.
2. The same Maxwell's equations for light propagation is used in all systems.
3. The length scale in each system is set according to the SI standard with the same speed of light in all systems in compatibility with 2.
4. All observers share the laws of Newtonian mechanics.
A2 as Assumption 2 means that each system carries its own "aether" as the coordinate system used to express Maxwell's equations, in accordance with this post. MMR is a "many-aethers"-theory with as many "aethers" as spatial coordinate systems. A2 means that the speed of light is the same in all systems, by SI standard.

A3 means that the same length scale is used in all systems, by SI standard.

A1 + A3 means that time and length scale is the same in all systems. There is no time dilation or space contraction in MMR.

In classical Newtonian/Galilean relativity there is "one-unique-aether", while in MMR there are many  (non-unique) aethers.

The reason Newtonian/Galilean relativity was replaced by SR was the Michelson-Morley experiment MM, which showed to be in conflict with a "one-unique-aether". On the other hand, the null result of  MM is in harmony with MMR, since each arm in the MM experiment carries its own coordinate system.

MMR is thus a "many-observer"-theory, where different observers use different coordinate systems moving with to respect to each other connected by the Galilean transformation, while they share time and length scale and the same Maxwell's equations expressed the same way in their respective systems.

Recall that SR shares the Lorentz transformation with a "Lorentz unique-aether"-theory, including time dilation and space contraction, with the difference that Lorentz considered these effects to be more illusion than reality. Time dilation and space contraction present a veritable night-mare to science with a wealth of "paradoxes" to handle, and so each possibility of getting out of the trauma must be tried.

But different observers of MMR may not agree on everything, such as Doppler shifts because Doppler shifts depend on the motion of source and receiver, and the observers are moving with respect to each other.

To see this consider a (1d) situation where at a specific time $X$ and $X^\prime$ with observers $O$ and $O^\prime$ in their respective coordinate origin overlap, while moving with velocity $w$ with respect to each other.  Consider the perceptions of a signal sent from a source at frequency 1, assuming the speed of light is normalised to 1, moving with velocity $v$ with respect to $X$. The frequency $f$ recorded by $O$ using $X$ will then be
• $f=\frac{1}{1+v}$
since the source is moving with speed $v$ (to the right say) and the receiver is fixed. On the other hand $X$ would attribute the following frequency to an observation made by $O^\prime$:
• $\bar f =\frac{1-w}{1+v}$ for $X^\prime$,
because $X$ sees the the receiver of $O^\prime$ moving with speed $w$ (to the left say). But the frequency $f^\prime$ de facto observed by $O^\prime$ will be
• $f^\prime =\frac{1}{1+v+w}$
since $v+w$ is the velocity of the source with respect to a fixed $O^\prime$. Computing we find that
• $\bar f - f^\prime =-\frac{w^2+vw}{(1+v)(1+v+w)}$,
which is second order in $v$ and $w$ compared to the speed of light = 1. Further aspects are given in  the MMR book.

The observers $O$ and $O^\prime$ will thus share the same time and length scale, but will have different perceptions depending on mutual motion, which can differ up the square of motion speed vs speed of light. With a speed of light of $3\times 10^8$ meter/second the difference would be of size $10^{-6}$ for velocities up to $300$ km/second. In short MMR appears to give a consistent description which can be shared by all observers up to second order accuracy in speeds. For observers on Earth $w$ may be up $1$ km/second and for observers in space up to $10$ km/second, far away from $300$ km/second.

We recall that the addition of two velocities $v$ and $w$ from composite Doppler shift in MMR reads:
• $v+w+vw$,
since
• $\frac{1}{1+w}\frac{1}{1+v}=\frac{1}{1+v+w+vw}$.
Altogether, MMR is compatible with the MM null result, and thus offers an alternative to SR which is compatible with Newtonian mechanics and electromagnetics (up to second order in relative light velocity).

The idea of a "many-aethers"-theory can be found in the work by the English mathematician Ebenezer Cunningham, but it was swamped by Einsteins "no-aether"-radicalism.

One can argue that the idea of a "many-aethers"-theory connects post-modern multiculturalism, while Einstein's "no-aether"-theory would correspond to a "no-culture-at-all".

PS1 Lee Smolin writes in Einstein's Lonely Path:
• Special relativity was the result of 10 years of intellectual struggle, yet Einstein had convinced himself it was wrong within two years of publishing it. He rejected his own theory, even before most physicists had come to accept it, for reasons that only he cared about. For another 10 years, as others in the world of physics slowly absorbed special relativity, Einstein pursued a lonely path away from it.
Einstein thus gave up SR in 1907. It is high time for physicists of today to do the same thing.

PS2 The physics community views Einstein with mixed feelings: Sometimes as a God but more often as a crackpot. One may argue that "once champ always champ", or "once crackpot always crackpot". What is your view?

PS3 Of course there is no twin paradox to resolve in MMR, since all twins age at the same rate according to the same clocks, independent of inertial motion. Of course acceleration or gravity forces or temperature, pressure and friction may affect the workings of a mechanical clock and thus clock rate (all clocks are mechanical), but certainly not pure translation with constant velocity, against the core of SR.

## onsdag 14 augusti 2019

### What is the Correct Resolution of the Twin Paradox if Any?

In discussions with theoretical physicists I have met the following "resolutions" of the twin paradox of special relativity SR as the paradox/apparent contradiction of different ageing of two twins, one staying at home and the other traveling on a round-trip (showing to be younger at return):
1. The different ageing shown in SR is real physics and can be explained within SR.
2. SR says nothing about the physics of ageing due to round-trip travel and so there is no twin paradox to resolve.
3. The different ageing shown by SR is real but can only be explained by the general theory of relativity.
Since 1.-3. are contradictory, I have asked a panel of theoretical physicists about the correct resolution, and will report the answers, when received. You are also welcome to submit your own resolution as a contribution to the discussion.

My question is the same as posed in an Open Letter signed by 142 physicists and others directed to the physics community, which received a very vague response. Read and contemplate! You will find clear evidence that there is no commonly accepted resolution, only resolutions which are viewed to be incorrect by parts of the community.

It reminds about the explanation of flight, for which the aerodynamics community only offers a number of different contradictory versions listed on e.g the NASA website as all incorrect, but no explanation claimed to be physically correct. For a physically correct explanation, see The Secret of Flight.

My experience so far (which is the same as that recorded in the Open Letter) is that leading physicists  are not on request willing/able to present a resolution of the twin paradox. What you get is:
• The paradox was solved very long ago in some way which no longer has any interest.
• Take a look at what wikipedia says.
• It can easily be solved within SR by using simple space-time diagrams.
• A resolution can be found by invoking general relativity, but that is so complicated that details cannot be given.
• The twin paradox is of interest only to crackpots, not to professional theoretical physicists who have many much more urgent questions to tackle.
What you don't get is anything claimed to be a correct solution accepted as such by the theoretical physics community. This is more than 100 years after the paradox was formulated.  If you think that what I say cannot be true, try out by asking the question yourself to your physics teacher or college.

The only way out in this hopeless situation is to opt for the "no paradox" version insisting that SR is logically consistent and as such free of paradoxes and beyond reach for physical paradoxes. But this would mean that SR is not a theory about some physics, which can (appear to) be paradoxical or be false, only a mathematical theory identical to the Lorentz coordinate transformation. The effect of such a step would be far-reaching, since modern physics is based on SR, and if SR is empty of physics that would make a lot of modern physics empty as well. So this way out is not possible...

PS If you repeat your question, because you don't get any reasonable answer, then you are met with anger and frustration, which is understandable if not very pleasant, or simply silence.

### Galilean Special Relativity as a Many-Aethers Theory

Einstein's special theory of relativity SR is based on the following postulates:
1. Laws of physics take the same form in all inertial systems.
2. The speed of light is the same in all inertial systems.
An inertial system is a Euclidean coordinate system in space together with a time coordinate and different inertial systems (with the same spatial orientation) are moving with constant velocity with respect to each other.

Consider the following realisation of 1+2: Assume
• Inertial systems are connected by a Galilean transformation.
• Newtonian mechanics.
• Maxwell's equations for electromagnetics in each inertial system.
Newtonian mechanics is Galilean invariant and thus satisfies 1. Assuming that identical Maxwell's equations are used in all inertial systems makes 1 true by definition, and then also 2.

Using formally identical Maxwell's equations in all inertial systems can be viewed as a "many-aethers"-theory with each Euclidean coordinate system representing an "aether" for propagation of light. The speed of light will then be the same in all systems. Compare with this post.

This is the set-up in Many-Minds Relativity as a "many-aethers"-theory. It should be compared with Einstein's "no-aether" SR, where Galilean transformation is replaced by Lorentz transformation.

A Galilean transformation connects the space coordinates $x$ and $x^\prime$ of two inertial system moving with velocity $v$ with respect to each other by the simple transformation
• $x^\prime = x-tv$
where $t$ is a time coordinate shared by both systems. A Galilean transformation has a direct simple realisation as a translation  with constant velocity, as the simplest possible motion.

A Lorentz transformation mixes space coordinates with time and has no physical realisation. Even more disturbing: Newtonian mechanics is not Lorentz invariant and thus has to be given up in SR. Einstein paid his tribute in his "Newton, forgive me!"

How could then Einstein end up with his SR based on Lorentz transformation with all its mysteries and sacking of Newton, starting from the same basic postulates 1+2  as we saw could as well be satisfied by Galilean transformation without mystery and with Newton intact?

The answer is hidden in Einstein's derivation of the Lorentz transformation from 1 + 2, which starts with light pulses initiated in two different inertial systems. By identifying the two light pulses to be one and the same taking different expressions in the two systems, Einstein then derived the Lorentz transformation. But the identification is unphysical in the sense that initialisation as coexistence at an initial time of a wave form in space necessarily takes different forms in different systems when space is mixed into time. It means that the identification of the pulses cannot be made and so Einstein's derivation of the Lorentz transformation from 1+2 is incorrect from physical point of view.

In his derivation of the Lorentz transformation Einstein relied on the concept of event  as something of unknown nature which has no extension in space, which can be labeled with a single space coordinate $x$ and time coordinate $t$. This made it possible for Einstein to view the launch of the  light pulses as one and the same event with different labels in the different systems and from that derive a connection between the labels in the form of Lorentz transformation.

An event labeled by $(x,t)$ carries unclear physics and can lead to misunderstanding of physics. The notion of particle as something without spatial extension but still physical presence, also can lead to misunderstanding.

Sum up: Einstein's SR is a "no-aether"-theory with strange physics in conflict with Newtonian mechanics. Many-Minds Relativity is a natural "many-aethers"-theory in harmony with Newtonian mechanics and electromagnetics.

PS To see the difference between one-pulse and two-pulse physics, consider two intertial systems which coincide at the light pulse launch. In the Galilean setting the launch physics will be the same in both systems which effectively means launch of two pulses with different translation speeds in two different "aethers", while in the Lorentzian setting they will be the same which is unphysical.

## söndag 11 augusti 2019

### Modern Physics in Free Fall Crisis

 Modern physicists in joint free fall under quantum supergravity.
There are many witnesses of a modern physics in serious crisis. The process started at the turn to 20th century modernity with Einstein's special theory of relativity and Planck's derivation of the law of blackbody radiation based on statistics of energy quanta opening to quantum mechanics.

Evidence of the crisis can be seen in the 2019 Special Breakthrough Prize in Fundamental Physics to a 1970 speculation about supergravity, which has resisted 50 years of experimental verification, see also Where are we now?. The Breakthrough website motivates the Prize to supergravity as follows:
• In the four decades since its development, supergravity has had a powerful influence on theoretical physics. It showed that supersymmetry was capable of accounting for all the phenomena we see in the real world, including gravity. It represented a completion of the current understanding of particle physics – a rigorous mathematical answer to the question, “What theories of nature are compatible with the principles of both quantum mechanics and special relativity?” And it provided a foundation for the attempt – still ongoing – to build a full theory of quantum gravity that describes space and time at a fundamental level.
We see that the 2019 Breakthrough Prize concerns precisely the question discussed in this sequence of posts: How to reconcile the principles of quantum mechanics and special relativity? But the 1970 answer in the form of supersymmetry appears to have few proponents today outside the Prize committee as a (failed) "attempt". It is not impossible that the 2020 Fundamental Physics Prize will go to the discovery that special relativity is not fundamental physics, and thus that there is no contradiction with quantum mechanics.

The shift to modernity was a break-off from classical physics as science of "what is" (ontology) into a modern physics as science of "what we can say" (epistemology) as expressed by Niels Bohr, in which a material world going around even without any (human) observer was replaced by a mist of statistics of (human) observation.

In the new view of modern physics causality/determinism was given up, under much agony because that had been the basic principle of physics since Aristotle, while the monumentality of the sacrifice added to its thrill.  But a sacrifice carries a cost and the cost is now showing up in the form of a modern physics in free fall without any thinkable connection to experiments as string theory in 11 dimensions and multiversa statistics of all possibilities.

What is then the effect of a physics in free fall? Is it helpful to humanity? What was the basic reason that forced Einstein and Bohr followed by generations of modern physicists, to give up causality and rationality?

Einstein was led to special relativity in an effort to handle the lack of physicality of a vacuum or "aether" as a medium for propagation of electromagnetic waves/light. It appeared in experiments like that by Michelson-Morley as if there was not just one single "aether", but many different as if each source/receiver system in motion would "drag" its own aether along. Einstein however could not handle the diversity of many aethers (put forward by e.g. Ebenezer Cunningham) and so took the radical step of declaring that there is no aether at all, in particular not many aethers causing confusing. In modern psycho-physiological terms it could be described as a syndrome of not being able to handle the many sometimes conflicting perspectives of life.

In any case the "no-aether" idea led Einstein into the his special relativity where all observers are compelled to share the same mathematical formulas under a banner of Lorentz invariance, however  without being able to agree on anything else of importance such as simultaneity, time and space. The scientific world met Einstein's special relativity with a yawn as epistemology without physics, which made Einstein turn to his general theory of relativity, which as a consequence of efforts to reconcile England and Germany after World War I in the hands of Eddington, took off in the media and then was turned into a pillar of modern physics.

The trouble was that this pillar was incompatible with the other pillar of modern physics, namely quantum mechanics founded on Schrödinger's equation, which resisted to Lorentz invariance. Modern physics has carried this incompatibility for 100 years as a basic trauma on which much of the present crisis can be blamed, with supergravity as failed attempt to reconcile quantum mechanics with gravitation. The rest of the blame can go to the use of statistics as a collapse of causality/determinism, which was forced from the multi-dimensional form of Schrödinger's equation.

Modern physicists bear witness of the crisis, but I have met little interest in possible ways to take off instead of falling down, by questioning Lorentz invariance and the necessity of a statistical approach to the physics of an atom. But the crisis goes on and maybe some day, discussion will be possible.

The similarity with a climate science dominated by one gospel of alarm is obvious. More precisely, the corruption of climate science today is made possible by the fact that modern physicists have retreated from the world.

The last sequence of posts give arguments that Lorentz invariance has no role to play in physics.
In Real Quantum Mechanics a variant of Schrödinger's equation free of statistics as a system in 3 space dimensions, is presented.

Comments from physicists are welcome. Is any form of discussion possible? Or is the crisis permanent?

PS In many respects modern physics appears as a game of poker, where the physicist player all the time can raise the bet and avoid being called by the public/tax payer. This is what Einstein did, when confronted with questioning of the special theory of relativity, by turning to general relativity, which when questioned was further raised to cosmology. Or when Schrödinger's equations for atoms was confronted with Lorentz invariance and the bet was lifted to Dirac's equation, not for atoms but only for one free electron, and then further to quantum field theory with a universe of infinities, which was handled by "renormalisation" and so on to string theory in 11 dimensions, which cannot be called because it is very far beyond both experimental conformation and refutation.

## lördag 10 augusti 2019

### The Seduction and Spell of the Lorentz Transformation

Modern physics is based on the idea of Lorentz invariance as the basic postulate of Einstein's special theory of relativity:
• Laws of physics are invariant under the Lorentz transformation between different inertial systems, that is, laws of physics are Lorentz invariant.
Recall that the Lorentz transformation connecting two inertial space-time coordinate systems $(x,t)$ and $(x^\prime ,t^\prime )$ for two observers moving with velocity $v$ with respect to each other, reads:
• $x^\prime =\gamma (x - vt)$, $t^\prime =\gamma (t - vx)$,
• $x =\gamma (x^\prime + vt^\prime )$, $t =\gamma (t^\prime + vx^\prime )$,
where $\gamma = \frac{1}{\sqrt{1-v^2}}$ assuming the speed of light is 1 and $\vert v\vert \lt 1$. We see that the space coordinate $x$ and time coordinate $t$ appear in symmetric form with a an apparent similarity between space and time, which Lorentz viewed to be a formality without physics, but Einstein took as a basis of modern physics with space mixed into time.

Which laws of physics are then formally Lorentz invariant? By the chain law, we have
• $\frac{\partial}{\partial x}=\gamma (\frac{\partial}{\partial x^\prime}-v\frac{\partial}{\partial t^\prime})$,
• $\frac{\partial}{\partial t}=\gamma (\frac{\partial}{\partial t^\prime}-v\frac{\partial}{\partial x^\prime})$,
and so
• $\frac{\partial}{\partial t}-\frac{\partial}{\partial x}=\gamma (1+v)(\frac{\partial}{\partial t^\prime}-\frac{\partial}{\partial x^\prime})$,
• $\frac{\partial}{\partial t}+\frac{\partial}{\partial x}=\gamma (1-v)(\frac{\partial}{\partial t^\prime}+\frac{\partial}{\partial x^\prime})$,
• $\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2}=(\frac{\partial}{\partial t}-\frac{\partial}{\partial x})(\frac{\partial}{\partial t}+\frac{\partial}{\partial x})$
• $=\gamma^2(1-v^2)(\frac{\partial}{\partial t^\prime}-\frac{\partial}{\partial x^\prime})(\frac{\partial}{\partial t^\prime}+\frac{\partial}{\partial x^\prime})=\frac{\partial^2}{\partial t^{\prime 2}} - \frac{\partial^2}{\partial x^{\prime 2}}$.
We see that the second order wave equation
• $\frac{\partial^2u}{\partial t^2} - \frac{\partial^2u}{\partial x^2}=0$,
is Lorentz formally invariant, in the sense of reading exactly the same in the $(x^\prime ,t^\prime )$ system.  On the other hand, for the first order wave equation:
• $\frac{\partial u}{\partial t} - \frac{\partial u}{\partial x}=0$,
the multiplicative factor $\gamma (1+v)$ appears, and so only a form of restricted formal Lorentz invariance is in place.

The second order wave equation describes waves in an elastic string with clear material spatial presence or coexistence, as well as plane electromagnetic waves in a vacuum without material spatial presence.

The idea of Einstein (picked up from Lorentz) was that since the wave equation takes the same form in all inertial systems connected by the Lorentz transformation (more or less), all inertial systems are equally valid (with in particular the same speed of wave propagation/light), which Einstein declared to be the essence of the new physics of the special theory of relativity. This was the seduction of Lorentz invariance which promised to solve the mystery on an "aether" medium for the propagation of electromagnetic waves in vacuum without material presence.

For both wave equations we see the space coordinate $x$ and time coordinate $t$ appearing in symmetric form, which opens up to some invariance with respect to the Lorentz transformation with similar symmetry.

But the formal symmetry in space and time in the wave equations does not say that space and time have the same nature and can be mixed into each other. In the wave equations there is a clear distinction between space and time which is expressed in the initial condition complementing the wave equation in a mathematical description of a wave $u(x,t)$, which takes the form
• $u(x,0)=u_0(x)$ for all $x$,
for the first order equation (with also an initial condition for $\frac{\partial u}{\partial t}$ in the second order case), where $t=0$ is an initial time and $u_0(x)$ an initial wave form with extension in space. With the initial condition a clear distinction between space and time is made. This is physics which is very obvious for the elastic string but also relevant for electromagnetic waves. The initial wave form shows spatial coexistence at different points $x$ for some common initial time $t=0$.

And now comes the catch showing that the Lorentz transformation is not compatible with physics, even if the wave equation is formally (more or less) Lorentz invariant: The initial condition is not invariant under Lorentz transformation, because $(x,0)$ translates into
• $(x^\prime ,t^\prime ) =\gamma (x, -vx)$,
which does not have the form of an initial condition for $t^\prime =0$. The physics of coexistence expressed in the $(x,t)$-coordinates through the initial condition for $t=0$ does not carry over to physics of coexistence of an initial condition for $t^\prime =0$. This means that the physics expressed in the different inertial systems is different. The whole idea of relativity of expressing the same physics in different inertial systems thus collapses.

The formal symmetry of the space and time coordinates in the two forms of the wave equation misled a confused Einstein to believe that space and time could be mixed, because Einstein did not properly understand the physical meaning of the mathematics of the wave equations. The sad fact is that generations of physicists have followed in the footsteps of Einstein with a mantra of Lorentz invariance as a necessary requirement of a law of physics.

A traveling wave is a solution $u(x,t)$ of either of the above wave equations of the form
• $u(x,t)=f(x+t)$,
where $f(\cdot )$ is a function of one variable. For example $f(y)=\sin(y)$ with
• $u(x,t)=\sin(x+t)$.
The initial condition for $t=0$ would then have the form $u_0(x)=\sin (x)$ as a wave in space, while an observer sitting at $x=0$ would experience a wave in time of the form $\sin(t)$, but the observer would have no reason to mix the wave in space with the wave in time just because the mathematics looks the same. To do that as Einstein did, shows that the mathematics is misunderstood.

The second order (but not the first order) equation also has standing wave solution of the form
• $u(x,t)=sin(t)sin(x)=\sin(\gamma (t^\prime +vx^\prime))\sin(\gamma (x^\prime +vt^\prime ))$,
with seemingly stationary spatial character in $(x,t)$-coordinates, but visibly not so in $(x^\prime ,t^\prime )$ coordinates. A standing wave solution is not Lorentz invariant. A standing wave for the observer using $(x,t)$-coordinates is not a standing wave for the observer using $(x^\prime ,t^\prime )$, of course not since the observers are moving with respect to each other.

In general the equations of mathematical physics do not show the symmetry of space and time of wave equations and thus do not show any Lorentz invariance at all. The physics is the same for all observers but its mathematical description varies between moving observers, as soon as the physics has some spatial presence, which is the nature of physics.  Only Maxwell's equations for vacuum can show formal Lorentz invariance, but not in the presence of charges and not with respect to initial conditions. The equations of physics are not Lorentz invariant. Not Maxwell with charges, not Schrödinger, not MHD, not Navier, not Navier-Stokes, not anything.

The net result is that the notion of Lorentz invariance has only a purely formal mathematical meaning and carries no real physics. If the spell of Lorentz invariance can be broken, then many possibilities  to progress seem to open up. But this is not something physicist like to hear. They will cling to Lorentz invariance no matter the cost and lack of reason. It is a spell.

Getting out of the spell means understanding that Leibniz distinction between space and time is valid also for modern physics:
• space = order of coexistence.
• time = order of succession.
But the spell has such strong grip on the minds of modern physicist that not even a basic discussion is possible.

## torsdag 8 augusti 2019

### Why so Difficult to Discuss Special Relativity with Physicists?

 I fooled you.
The attempted discussion with Swedish media physicist Ulf Danielson illustrates the difficulties met when trying to discuss aspects of the special theory of relativity SR with physicists, who must defend SR by all means to keep their academic jobs. The difficulties arise because, in a tradition mastered by Einstein, it is never clear what the premises are. In particular, there are several forms of SR:
• SR = consequences of the two postulates of SR. (1. Laws of physics must take the same form in all inertial systems. 2. Measurements of the speed of light must give the same result in all inertial systems).
• SR+ = SR + some physics.
• SR++ = SR+ + some more physics.
The postulates of SR have the form of stipulations without concrete physical content and the Lorentz transformation derived from the postulates is a coordinate transformation between inertial systems, which as such has no concrete physical content. Postulate 1 says that laws of physics must take the same form in all inertial systems, that is they must be Lorentz invariant. SR is thus empty of real physics as a specific coordinate transformation, which has no truth value as being the result of stipulations without concrete physics. Einstein's used the term Postulate to signify that SR is based on stipulations without truth value and not propositions about physics, which may be true or false. Since Postulates 1 and 2 are not contradictory, a physicist can argue that SR is not contradictory.

On the other hand, one can argue that Postulates 2  is a consequence of Postulate 1 and as such redundant, but not contradictory.

SR+ = SR + Maxwell's equation for electromagnetic waves in a vacuum can be argued to be correct physics because Maxwell's equations for vacuum describe physics and are Lorentz invariant. But SR does not add anything to such a picture and can then as well be discarded.

SR+ with physics in the form of clocks and traveling symmetric twins with unsymmetric ageing leads to a paradox, which is "resolved" by adding some more physics like one twin staying at rest at home and the other traveling undergoing acceleration and retardation into a S++ with unsymmetric ageing seemingly without paradox.

In this way SR+ can be just anything and as such can be twisted to fit with observations with the message that SR+ has experimental support including the postulates of SR, which in fact are stipulations beyond experimental verification. By adding some physics some experimental support can be constructed, but the role of SR itself is then unclear since the support concerns the added physics.

A physicist can thus ague that SR in basic form contains no contradiction, and in suitable SR+ form has experimental support. The message is that SR is a theory without contradiction, which has massive experimental support. It could not be better. With this message the physicist can keep his job.

But there is one form of SR+ which shows to be incompatible with experiments, and that is SR+ = SR+gravitation, which was the dream Einstein had to abandon as soon as SR was confronted with the reality of gravitational forces. A desperate Einstein tried to save his scientific life by cooking up a form of SR including gravitation, by effectively reducing to SR without gravitation by sweeping gravitational forces/acceleration away into geometry, named general relativity. Einstein thus gave up the basic stipulation of SR of Lorentz invariance to keep his job. The question is why today physicists have to in public confess to the sermon of Lorentz invariance to keep their jobs: When laws of gravitation are not Lorentz invariant. When Schrödinger's equation of quantum mechanics is not Lorentz invariant. When the equations describing the mechanics of solids and fluids are not Lorentz invariant. When Maxwell's equations with charges are not Lorentz invariant. Why cling to Lorentz invariance when doing so misses the whole cake?

Summary: SR + Maxwell in vacuum is fine, but here SR has no role since Maxwell in vacuum carries itself. SR + gravitation does not work. The net result is that SR can be put into the wastebin of thought experiments without physics, and when that is done a step forward can be taken. Too long has physics been forced into a straitjacket of Lorentz invariance invented by the young ambitious Einstein from which the mature Einstein managed to escape but not his followers.

### Why the Speed of Light Can Be Independent of Translation of Source-Receiver

To see that it is most natural that the speed of light between a source and receiver is independent of translation with constant velocity of both source and receiver, as if the "aether" was moving along with the same velocity, consider the following mechanical analog:

Connect a source to a receiver by waves carried by a stretched rope with the source at one end and the receiver at the other end, and assume that the whole system source-rope-receiver is subject to a translation with constant velocity. We would then make the observation that the speed of propagation of the waves from source to receiver through the rope would be independent of the translation.

This is because the rope as the medium for wave propagation moves along with the source/receiver. The analog would be an "aether" as an immaterial medium for propagation of light waves, which moves along with the source/receiver. The idea is that the light source and receiver establishes a connection like a rope which is moving along with the source/receiver.  This would be compatible with the null result of the Michelson-Morely experiment.

There is thus no common background aether, but a multitude of aethers connecting and moving along with a multitude of sources/receivers.  The immaterial quality of aethers would then allow a multitude without conflict. This is the idea of Many-Minds Relativity. Think of that.

PS If the source and receiver move with respect to each other they could still share a common aether (connected to either source or receiver) with still  he same speed of light, while light frequencies would be subject to Doppler effects.

### Special Theory of Relativity Incompatible with Gravitation

 Sorry to say, but I had to abandon my most beautiful special theory of relativity SR because it could not be combined with laws of gravitation. The demand of Lorentz invariance of course was not compatible with the presence of matter in space. I thus turned to the general theory of relativity giving up Lorentz invariance for covariance, and never returned to SR.

From its start in the 1905 article On the Electrodynamics of Moving Bodies, Einstein connected his special theory of relativity SR to Maxwell's equations for electromagnetic waves. From the two Postulates of SR (1. Relativity Postulate and 2. Constancy of speed of light). Einstein derived the Lorentz transformation connecting coordinates in different inertial systems moving with constant velocity with respect to each other. The catch was that Maxwell's equations in vacuum showed to take the same form in all coordinate systems connected by the Lorentz transformation and thus showed to be Lorentz invariant as required by the Relativity Postulate.

Einstein then tried to extend SR with its requirement of Lorentz invariance to gravitation, but failed and so Einstein abandoned SR for his general theory of relativity. The situation is the same today:
• SR cannot be extended to include gravitation.
• Laws of gravitation are not Lorentz invariant.
This is shown in e.g. Relativistic Theories of Gravitation by Withrow and Murdoch giving a survey of attempts to extend SR to gravitation,  all failing because of incompatibility with observations. The trauma of modern physics can thus be captured as follows:
• Newton's mechanics is Galilean invariant but not Lorentz invariant.
• Maxwell's equations are not Galilean invariant, but Lorentz invariant (in a restricted sense).
• SR is incompatible with mechanics including quantum mechanics.
The trauma is a result of insisting on the Relativity Postulate asking a "law of physics" to have the same formal expression in all inertial systems connected by the Lorentz transformation, that is to require Lorentz invariance.

What then to do?
1. Allow the trauma to continue to paralyse modern physics?
2. Give up Lorentz invariance as a necessity to impose on laws of physics?
I suggest 2. This means that SR is given up because it is incompatible with in particular laws of gravitation. Laws of gravitation cannot be Lorentz invariant if they are going to match observations. This was what Einstein understood when turning to his general theory of relativity. If a theory does not fit with experiments, like SR with gravitation, then it has to be given up.

What then would be the effect of giving up SR? Nothing! It is illustrated by the fact that GPS system works because SR is not included, not because SR is included.

What then about the null result of Michelson-Morley experiment, as the main motivation for asking for Lorentz invariance? Is it possible to explain the null-result without invoking Lorentz invariance? Of course, as will be shown in an upcoming post.

The reason Maxwell's equations in vacuum can be viewed to be formally Lorentz invariant, that is take the same in all inertial frames, is that the vacuum has no material presence in space. As soon as you introduce material presence in space in the form of charges or matter, the idea of Lorentz invariance collapses because necessarily the spatial motion of the inertial system vs the charges/matter must come in. Since SR is incompatible with any form of material/spatial presence, it has to be given up, as Einstein did. Galilean invariance is meaningful, but not Lorentzian.

I have argued that the two postulates of SR themselves are empty of physics. It is possible to give SR a restricted physical meaning by combining the postulates with Maxwell's equations in vacuum showing some compatibility with experiments.  But adding physics in the form of gravitation has shown to not be compatible with experiments. It is now time also for us to give up SR, following Einstein.

Or restrict SR to the domain of its postulates without combination with any physics, in which case SR is empty of physics and carries no incompatibility with observation of physics.

PS In the debate with Ulf Danielson in a previous post, it was not made clear what physics was appended to the two postulates of SR themselves empty of physics, which led to a lot of confusion. Let's see if UD will take up the challenge and comment this post...

## tisdag 6 augusti 2019

### Does the Period of a Harmonic Oscillator Change under Uniform Translation?

 High speed trains in China in uniform translation with identical clocks.
The most basic of all clocks is a harmonic oscillator consisting of a body of unit mass connected by a linear spring to a fixed point, described by the differential equation expressing Newton's 2nd Law:
• $\frac{d^2x}{dt^2}=(x_0-x)$   (1)
where $x(t)$ is the position of the body at time $t$ on an $x-axis$ and $x_0$ is the fixed point with $x_0-x$ the spring force. The solution $x(t)$ is periodic in time with period $2\pi$.

Suppose now we introduce another coordinate axis with coordinate $x^\prime = x-vt$, where $v$ is a given constant velocity, expressing that the $x^\prime$-axis moves with the constant velocity $v$ with respect to the $x$-axis in uniform translation. Substituting $x=x^\prime +vt$ into (1), we get
• $\frac{d^2x^\prime}{dt^2}=(x_0^\prime-x^\prime )$   (2),
because $\frac{d^2(vt)}{dt^2}=0$. We see that (1) reads identically the same as (2) sharing the same time $t$. We thus see that the motion of a harmonic oscillator is Galilean invariant since the equation describing the motion reads the same in the two space coordinate systems connected by the Galilean transformation $x^\prime = x -vt$ of uniform translation.

One way to express our experience is to say that the motion of a harmonic oscillator including the period is independent of uniform translation. This is what we expect: It is unthinkable that the physics including the period of a harmonic oscillator, could be influenced by uniform translation. Your clock must tick the same rate waiting for the train in the train station and in the train in uniform translation. Anything else is unthinkable, based on the basic Newtonian mechanics of a harmonic oscillator.

But the special theory of relativity SR says that the clock in motion ticks at a slower rate.

What is your conclusion? Evidence that SR is correct or false from a physical point of view?

PS1 Note that (1) has the same form in all inertial systems under uniform translation and thus satisfies Einstein's Relativity Postulate. What is then wrong with (1) from the view of SR? Well, SR is obsessed with the speed of light, but a harmonic oscillator has nothing to do with light because it is a mass-spring system. So what is wrong with the harmonic oscillator form the point of view of SR is that it is mechanical, but is it anything wrong with being a mechanical system?  Isn't a harmonic oscillator an example of basic physics?  If not, what physics is then SR?

PS2 To see that (1) is not Lorentz invariant as required in SR, recall this post. It means that (1) from the point of view of SR is not a law of physics. Einstein thus claims that a harmonic oscillator is not physics.  Do you agree?

PS3 Things like this appears to be impossible to discuss with physicists seemingly brain-washed by special relativity.

### Special Relativity Theory for One Observer?

 A One-King system can be very stable.
Einstein's special theory of relativity concerns the relation between observations (of "events" without spatial extension recorded by space-time coordinates (x,t)) made by different observers using different inertial coordinate systems moving with constant velocity with respect to each other.

Suppose there is only one observer making observations in only one coordinate system in which the observer is at rest. Let us refer to this as a one-system situation. Can there be any special relativity theory for this one-system with its single observer/inertial system? Of course not. With only one  inertial system there is no room for comparison with observations in another moving inertial system. The special theory of relativity is empty for a one-system.

Now, the GPS system is a one-system based on the WGS84 spherical (ellipsoidal) coordinates system we are used to with latitude, longitude and height for spatial location, and an Earth based master clock setting common time, which is what your GPS receiver can report to you. The GPS system is thus a one-system and as such has no use for the special theory of relativity. In this one-system, the position of moving satellites are recorded by one master observer at rest on Earth and satellite clocks are synchronized with the master clock to UTC Universal Central Time (with nanosecond precision). There are no observers, sitting in the satellites making observations in moving inertial systems requiring coordination using the special theory of relativity. This is a one-master one-system for which the special theory of relativity has nothing to contribute.

This is yet another observation that GPS does not depend on the special theory of relativity.

PS We may compare with Deng Xiaopin's idea of one country - two systems, which we now see is collapsing in Hong-Kong. A one-system appears to be more stable, which is also what we expect from GPS.

## måndag 5 augusti 2019

### Does GPS depend on Theory of Relativity?

The story goes that without taking both the special and general theory of relativity into account, the Global Positioning System GPS would not work: It would be off by 11 km over a day. This is stated in e.g. Stephen Hawking's bestseller A Brief History of Time, and in some books about GPS, but not all. In particular, the Official U.S. government web site about the GPS and related topics, tells nothing about the use of relativity theory. Is US hiding something?

So what are the facts? The satellite clocks all use the same cesium atomic clocks, which at satellite launch are slowed down the fixed amount of 38.000 nanoseconds per day, which is about 0.5 nanoseconds per second.

Some sources present the offset as a tribute to both the special (+ 7.000) and the general theory of relativity (-45.000), but not all and not the Official U.S. Government GPS web site.

Other sources say that the initial offset is made because the environment of the satellite clock makes them tick a little faster than the master ground clock, not because of relativity but because the mechanics of the clock is influenced by temperature, pressure and gravitation. In any case, the satellite clocks are continuously synchronized to an Earth based master clock to within 1 nanosecond per second allowing to set Coordinated Universal Time UTC with that precision.

From this information alone, we understand that the initial offset of 0.5 nanoseconds per second plays no role whatever the reason for it may be: Even without initial offset the system would work fine by the continuous synchronization used.

Let us now see what effect 1 nanosecond per second, that is a relative precision of $10^{-9}$, can have on the precision of the system. We know that both the position and clock reading of a GPS satellite when sending a signal received by the GPS receiver in your hand, is encoded in the signal. With simultaneous readings from 4 satellites it allows the receiver to both synchronize its own clock with the synchronized satellite clocks and from that determine the signal transit time and then the distance to the different satellites and then finally its own position. The travel time of the signal is less than 0.1 seconds, and so the effect on position from a 1 nanosecond per second time offset is $10^{-10}$, which with a speed of light of $3\times 10^8$ meter per second, would be 3 centimeters. In any case the effect of a difference of 1 nanosecond in travel time of light is 3 decimeter = 1 foot as possible GPS precision.

From where does then Hawking pick his 11 km/dag? Well, it comes from multiplying $38.000\times 10^{-9}$ seconds/day with the speed of light $3\times 10^5$ km/second to get
$11=3.8\times 3$ km/day. But this has nothing to do with reality. It is deceptive fiction propagated by physicists like Clifford Will to sell the message that Einstein's theory of relativity has massive experimental support.

Why do physicists deceive us with fake-physics? How can 3 centi/decimeters be twisted into 11 km?

## fredag 2 augusti 2019

### The Special Theory of Relativity = Pseudo-Physics?

 White Rabbit in Alice in Wonderland: I don't know what time is, but I can measure it using my cesium clock.
In the last sequence of posts the question is raised if Einstein's Special Theory of Relativity SR is empty of real physics. The answer must be hidden in the two basic Postulates of SR:
• 1. Laws of physics take the same form in all inertial systems (Relativity Postulate).
• 2. The speed of light is the same in all inertial systems.
Postulate 1 stipulates  that a "law of physics" must have a certain property, which is a ridiculus requirement sorting out almost all physical laws as we know them, but nevertheless does not itself express any real physics. Postulate 1 is thus empty of real physics.

Postulate 2 is according to the SI 1983 standard a definition of the length standard meter by stipulating that the speed of light is exactly 299792458 meter per second. A theoretical physicist claiming that SR is a theory about real physics, would say that Postulate 2 is both a definition/standard/convention/stiplulation and a statement about real physics in the sense that it expresses the fact that the speed of light "is" the same in all inertial systems. Both a definition/standard and physical fact. The claim is that standard works because the speed of light "is" constant, not only measured to be constant by definition. But is this reasonable?

To get perspective, let us make a comparison with the SI standard for measure of time as second defined by the
• unperturbed ground-state hyperfine transition frequency of the cesium-133 atom as 9,192,631,770 Hz or 1/second.
All observers following the SI standard will thus measure time with identical unperturbed (no exterior influence) cesium clocks. The rate of time will thus be the same for all observers (compare speed of light being the same for all observers in different inertial systems), by definition/standard.

Would it now be meaningful to claim as a Postulate saying something about real physics that the SI time standard works, because the rate of time "is the same" as some form of deep physics, which can be expressed in the following analog of Postulate 2:
• 3. The rate of time is the same in all inertial systems.
To me Postulate 3 does not make sense, because rate of time is what is measured by a cesium clock, by definition. It is meaningless to insist that the rate of time (or speed of light)  "is constant" in itself, when the fact is that it is a matter of measurement standard.  Speed of light and rate of time are derived entities depending on a standard and cannot be attributed having independent status as "being constant". Further, to "be constant" appears as antithesis of relativity.

This arguments indicates that Einstein's Postulate 2 does not make sense from a physical point of view. With the 1983 SI standard it is effectively only a definition void of real physics. It is a stipulation or prescription without physical truth content.

As with all standards, in particular for meter and second, the real question is to what degree different observers under different conditions following the same standard, will agree. This is discussed in the book Many-Minds Relativity. The idea is to chose the standard so that it can be shared by many. In particular, a cesium clock can be expected to show no dependence on choice of inertial system, and only very weak dependence on exterior conditions, but there may be a small dependence on e.g. temperature and gravitation like for a pendulum clock, which has nothing to do with relativity theory.

How well does the time standard work? Pretty good as evidenced by the amazing functionality and precision of the GPS system: All satellite clocks tick at the same rate synchronised with an Earth based reference clock, without being subject to any time dilation by different motion as predicted by SR. GPS works because SR is pseudo-physics, not because SR is real physics as claimed by many physicists as experimental support of SR.

We are led to conclude that both Postulate 1 and 2 of SR are empty of real physics, and so SR can only by pseudo-physics, not real physics.

PS1 Note that the SI standard specifies how to "measure time" (with an unperturbed cesium clock) as an operational definition, while the question what "time is" and even more specifically what the "rate of time is", can be left out.  This is a relief, since nobody has an answer.

PS2  To believe that something is true while not claiming real understanding,  which is the standard attitude vs SR, is one thing. Another thing is to understand that something is fundamentally wrong, which is the non-standard view on SR, which requires more insight than just believing.


text:
How does the teleportation circuit work after the initial entanglement?

I'm trying to understand why we do what we do when we teleport a state from Alice's first qubit to Bob's qubit. I understand the initial entanglement, but could use help walking through the program.

I have a teleportation circuit,

circuit = QuantumCircuit(3,3)

First, I create the bell pair for Alice and Bob:

circuit.h(1)
circuit.cx(1,2)

Then, I flip alices first qubit to a 1 state; this is the qubit whose state is to be teleported:

circuit.x(0)

Then, my understanding gets fuzzy, and I have a few questions.

1) I seem to entangle Alice's to-be-teleported state qubit with her qubit that is entangled with Bob's. But what exactly am I doing when I conditionally flip this to-be-teleported qubit and then put a hadamard on it?

circuit.cx(0,1)
circuit.h(0)

and then 2) how does my measurement affect Bob's qubit?

circuit.measure([0,1],[0,1])

finally, 3) why do I again conditionally flip Alice's second qubit with Bob's qubit and then conditionally rotate Alice's to-be-teleported qubit with Bob's qubit?

circuit.cx(1,2)
circuit.cz(0,2)

Here is the whole circuit, for reference:

format: plain


text:
# Math Help - show that V(X) = lambda

1. ## show that V(X) = lambda

I have a problem and I don't really want the solution just a push in the right direction.

given that $\Pr(X=r) = \frac{e^{\lambda}\cdot{\lambda ^{r}}}{r!}$ r = 0,1,.....

show that Varience of X equals $\lambda$.

2. Hello,
Originally Posted by Animalxxv
I have a problem and I don't really want the solution just a push in the right direction.

given that P(X=r) = $\frac{e^{lambda}\cdot{lambda ^{r}}}{r!}$ r = 0,1,.....

show that Varience of X equals lambda.

Note that $\mathbb{E}(f(X))=\sum_{r=0}^\infty \mathbb{P}(X=r) f(r)$
where $\mathbb{E}$ is the expectation.

We know that the variance V is given by this formula :
$V(X)=\mathbb{E}(X^2)-\mathbb{E}(X)^2$

$V(X)=\sum_{r=0}^\infty \mathbb{P}(X=r) r^2-\left(\sum_{r=0}^\infty \mathbb{P}(X=r)r\right)^2$

Can you do it ?

Edit : there are other ways such as using generating functions. This was the most basic one that came in my mind ><

3. Yep thank you very much for your help

4. Originally Posted by Animalxxv
I have a problem and I don't really want the solution just a push in the right direction.

given that $\Pr(X=r) = \frac{e^{\lambda}\cdot{\lambda ^{r}}}{r!}$ r = 0,1,.....

show that Varience of X equals $\lambda$.


text:
# List of short problem names?

I typeset a book with problems and solutions. I need to get a list of short problem names. I expected there will be a way similar to pitcures:

\caption[Figure name for List of Figures]{Caption name}

below is a summary of ways to get the list of problems.

# ntheorem

This approach seems to be closer than others:

\documentclass{book}

\usepackage{ntheorem}
\theoremlisttype{all}
\newtheorem{Problem}{Problem}[chapter]

\begin{document}

\listtheorems{Problem}

\chapter{foo}

\begin{Problem}[Short foo name]
Foo!
\end{Problem}

\chapter{bar}

\begin{Problem}[Short bar name]
Bar!
\end{Problem}

\end{document}

• The problem here is that the short name is also shown in the problem. Perhaps it is possible to define a new \newtheoremlisttype...

# thmtools

Here's a MWE thmtools (from http://tex.stackexchange.com/a/2500/8992)

\documentclass{book}

\usepackage{amsthm}
\usepackage{thmtools}
\declaretheorem[numberwithin=chapter]{hypothesis}
\declaretheorem[numberwithin=chapter]{theorem}

\begin{document}

\renewcommand{\listtheoremname}{List of Hypotheses}
\listoftheorems[ignoreall,show={hypothesis}]

\chapter{foo}

\begin{hypothesis}[X-Y-relationship]
There is a positive relationship between X and Y.
\end{hypothesis}

\begin{theorem}
Some text.
\end{theorem}

\end{document}

• There's no way to specify a name for a list of problems

# Hack

Here's MWE of some hack (from http://tex.stackexchange.com/a/6917/8992):

\documentclass{book}

\usepackage{ntheorem}
\newtheorem{hypo}{Hypothesis}[chapter]
\newtheorem{hypolist}{Hypothesis}[chapter]

\newcommand*\hypothesis[1]{%
\stepcounter{hypolist}%
\addtheoremline{hypolist}{#1}%
\begin{hypo}#1\end{hypo}
}

\begin{document}

\chapter*{List of Hypotheses}
\listtheorems{hypolist}

\chapter{foo}

\hypothesis{There is a positive relationship between $X$ and $Y$.}

\chapter{bar}

\hypothesis{There is a positive relationship between $X$ and $Z$.}
\hypothesis{There is a positive relationship between $Z$ and $Y$.}

\end{document}

• there's no way to specify a shorter name for the list of problems
-

## 2 Answers

You can simply redefine the plain theorem style to ignore the "attribution" part. The default definition is

\newtheoremstyle{plain}
{\item[\hskip\labelsep \theorem@headerfont ##1\ ##2\theorem@separator]}%
{\item[\hskip\labelsep \theorem@headerfont ##1\ ##2\ (##3)\theorem@separator]}

so you can say

\documentclass{book}

\usepackage{ntheorem}
\theoremlisttype{all}

\makeatletter
\newtheoremstyle{problem}
{\item[\hskip\labelsep \theorem@headerfont ##1\ ##2\theorem@separator]}%
{\item[\hskip\labelsep \theorem@headerfont ##1\ ##2\theorem@separator]}
\makeatother
\theoremstyle{problem}
\newtheorem{Problem}{Problem}[chapter]

\begin{document}

\chapter*{List of problems}
\listtheorems{Problem}

\chapter{foo}

\begin{Problem}[Short foo name]
Foo!
\end{Problem}

\chapter{bar}

\begin{Problem}[Short bar name]
Bar!
\end{Problem}

\end{document}

This is the list:

and here is the first problem text

-
How did You get the plain style definition? I've made it a separate question. –  Adobe Dec 10 '12 at 10:35
@Adobe I looked in the code of ntheorem.sty, searching for \newtheoremstyle. –  egreg Dec 10 '12 at 10:36
How do I control space between number and short problem tittle in the list of problems? I tried the following: \hskip\labelsep \theorem@headerfont ##1\ \hspace{1in} ##2\theorem@separator -- it compiles but doesn't work. –  Adobe Feb 17 '13 at 20:13
Here's the closest I can get: \makeatletter\def\thm@@thmline#1#2{\@dottedtocline{-2}{0em}{5em}{\protect\numberline{#2}}}\makeatother (here I'm powered solely by trial-and-error). –  Adobe Feb 17 '13 at 20:30
For the spacing, see tex.stackexchange.com/questions/53353/… –  egreg Feb 17 '13 at 20:50

For this type of situation, I'd be tempted to defined my own environment rather than hack into the internals of an existing package.

In the code below, I've defined two environments for you to play with, depending on what you want. Both approaches rely on the etoolbox for a few commands- the second environment needs xparse for an environment with two optional arguments.

Both environments add to \jobname.prb which is created using the \@starttoc command.

Problem

This environment takes an optional argument, which will be added to .prb if not empty.

% define the Problem environment
%
% Takes an optional description- if it is not empty,
% then it is added to \jobname.prb
\newcounter{problem}
\newenvironment{problem}[1][]{\refstepcounter{problem}%
{\bfseries Problem~\theproblem} %
\ifstrempty{#1}%
{%
% if #1 is empty, do nothing
}%
{%
% otherwise add it to jobname.prb
\addcontentsline{prb}{subsection}{#1}%
}%
}{}

This can be used as either

\begin{problem}
\lipsum[2]
\end{problem}

or

\begin{problem}[Description goes here, and added to jobname.prb]
\lipsum[2]
\end{problem}

Otherproblem

This one is a bit more intricate as it takes two optional arguments.

% define another Problem environment, just for demonstration
%
% Takes TWO optional arguments:
%   #1: long description, used for \jobname.prb if #2 is not present
%   #2: short description, used for \jobname.prb if present
\newcounter{otherproblem}
\NewDocumentEnvironment{otherproblem}{O{} O{}}{\refstepcounter{otherproblem}%
{\bfseries Problem~\theotherproblem} %
\ifstrempty{#2}%
{%
% if #2 is empty, then check for #1, and
% use it for jobname.prb if present
\ifstrempty{#1}%
{%
% if #1 is empty, do nothing
}%
{%
% otherwise, display it, and add it to
% \jobname.prb
{[\bfseries #1]}%
\addcontentsline{prb}{subsection}{#1}%
}%
}%
{%
% if #2 is not empty, use it for the
% \jobname.prb
\addcontentsline{prb}{subsection}{#2}%
\ifstrempty{#1}%
{}%
{%
{[\bfseries #1]}%
}%
}%
}{}

Sample usage

\begin{otherproblem}[Long description][Short description for list]
\lipsum[2]
\end{otherproblem}

or

\begin{otherproblem}[Long description displayed and added to list]
\lipsum[2]
\end{otherproblem}

or

\begin{otherproblem}[][Short description for list, nothing displayed]
\lipsum[2]
\end{otherproblem}

Here's the complete MWE to play with- let me know if you need help changing the style of the environments.

\documentclass{article}
\usepackage{lipsum}% just to generate text
\usepackage{etoolbox}
\usepackage{xparse}

% this sets up \jobname.prb which will store the
% contentslines added on each problem
\makeatletter
\newcommand\listproblemname{List of problems}
\newcommand\listofproblems{%
\section*{\listproblemname}\@starttoc{prb}}
\makeatother

% define the Problem environment
%
% Takes an optional description- if it is not empty,
% then it is added to \jobname.prb
\newcounter{problem}
\newenvironment{problem}[1][]{\refstepcounter{problem}%
{\bfseries Problem~\theproblem} %
\ifstrempty{#1}%
{%
% if #1 is empty, do nothing
}%
{%
% otherwise add it to jobname.prb
\addcontentsline{prb}{subsection}{#1}%
}%
}{}

% define another Problem environment, just for demonstration
%
% Takes TWO optional arguments:
%   #1: long description, used for \jobname.prb if #2 is not present
%   #2: short description, used for \jobname.prb if present
\newcounter{otherproblem}
\NewDocumentEnvironment{otherproblem}{O{} O{}}{\refstepcounter{otherproblem}%
{\bfseries Problem~\theotherproblem} %
\ifstrempty{#2}%
{%
% if #2 is empty, then check for #1, and
% use it for jobname.prb if present
\ifstrempty{#1}%
{%
% if #1 is empty, do nothing
}%
{%
% otherwise, display it, and add it to
% \jobname.prb
{[\bfseries #1]}%
\addcontentsline{prb}{subsection}{#1}%
}%
}%
{%
% if #2 is not empty, use it for the
% \jobname.prb
\addcontentsline{prb}{subsection}{#2}%
\ifstrempty{#1}%
{}%
{%
{[\bfseries #1]}%
}%
}%
}{}

\begin{document}

\listofproblems

\clearpage

\begin{problem}[Regular problem environment]
\lipsum[1]
\end{problem}

\begin{otherproblem}[Long description][Short description for list]
\lipsum[2]
\end{otherproblem}

\begin{problem}
\lipsum[2]
\end{problem}

\begin{problem}
\lipsum[2]
\end{problem}

\end{document}

-
I'll find time and study Your solution thoughly. –  Adobe Dec 10 '12 at 12:42


text:
# Introduction

This will take you through the basics of performing a 2D Plot in MATLAB.

# The figure Function

The function

figure

Will create a new figure using the lowest unused positive integer. e.g. if no additional windows were previously open it will create Figure 1.

If an input argument is selected. e.g.

figure(100)

MATLAB will instead create Figure 100.

If an input argument is selected of a figure that exists. e.g.

figure(1)

Then MATLAB will reselect Figure 1. Any sub-sequential commands will apply to that Figure.

# Getting the Data From Excel

## Copying Each Column to a New Variable

We have the data we want to plot in an Excel Spreadsheet. Although it is pretty easy to create Variables by typing it in, in MATLAB we will create a new variable call it v and copy the first column to it.

We will then do the same with the second column but this time calling it v:

We can dock Figure(1) by pressing the Dock Button:

I want it snapped to the right of Variables, not below them so I will drag it to the right:

Now the figure is docked:

Since I have x and y data in the form of t and v I can use the plot command:

plot(t,v)

We can also select the Columns we want to plot from a Variable that is a Matrix or alternatively if or input data is in the form of a Table we can select the Table Variables (Columns) to Plot.

## Copying All Columns to a New Variable and Selecting which Columns to Plot

If instead of copying the Excel Data to separate Variables t and v, it was copied over as a single Variable vt:

The Command

plot(tv)

Would give:

Note the previous plot in Figure 1 is overridden because Figure 1 is still selected.

In this dataset, no x values were selected so the row index was used for the x values and t and v were plotted with respect to this index.

This isn’t what we wanted, we wanted to plot the first column of vt which is t with respect to the second column of vt which is v. We can explicitly state this in the command:

plot(tv(:,1),tv(:,2))

## Making a Table from

For more practice we can also create a table from the columns v and t and plot by selecting the table variables

tabletv=table(t,v)

plot(tabletv.t,table2.v)

tabletv2=table(tv(:,1),tv(:,2))

tabletv2.Properties.VariableNames={'t','v'}

plot(tv2(:,1),tv2(:,2))

For the Plot Function, many of the additional input arguments come in pairs. The first value selects the setting to be changed and the second value selects the value for such a setting to be changed to. Unlike the names of functions or variables when the inputs are characters, they are not case sensitive.

## LineWidth

These input arguments specify the thickness of the line. The first value is a string of characters and the second value is a scalar.

plot(t,v,'LineWidth',5)

## Color

This is spelt using English (U.S.). The English (U.K.) spelling will not work.

These input arguments specify the colour of the line. The first value is a string of characters and the second value for primary and secondary colours can either be a string of characters as the full colour name or it’s abbreviation. Alternatively it can be specified as a [R,G,B] value. For this the RGB values are normalised between 0 and 1. In many other applications they range between 0 and 255.

plot(t,v,'Color','Red')

plot(t,v,'Color','R')

plot(t,v,'Color',[1,0,0])

All colours are made from a mixture of three primary colours:

• red
• green
• blue

Black is shown in the absence of these three colours and white is shown when these three colours are at full brightness.

Full NameAbbreviationRGB Value
'Red''R'[1,0,0]
'Green''G'[0,1,0]
'Blue''B'[0,0,1]
'Magenta''M'[1,0,1]
'Yellow''Y'[1,1,0]
'Cyan''C'[0,1,1]
'Black''K'[0,0,0]
'White''W'[1,1,1]

As seen a colour can be expressed as a [r,g,b] row vector opposed to a pre-existing string.

Many programs such as Microsoft Office list colors in 8 bit, which means they list 2^8=256 levels. If we subtract 1 from 256 to get a 0 value with range from 0-255. We may use the colour pickers in Microsoft Word to determine the colour we want to use. Select the button to change the colour of the text, select More Colours:

Select your colour and then select custom:

You now get your RGB values listed from 0 to 255.

We can create a row vector to it and assign it to a variable name. For instance using the three numbers shown above and dividing them through by 255.

mw_wine=[192/255,0/255,0/255]

Repeating this with the standard colours in Microsoft Word we get:

Custom Variable NameRGB Value
mw_wine[192/255,0/255,0/255]
mw_red[255/255,0/255,0/255]
mw_orange[255/255,192/255,0/255]
mw_yellow[255/255,192/255,11/255]
mw_lime[146/255,208/255,80/255]
mw_green[0/255,176/255,80/255]
mw_sky[3/255,177/255,240/255]
mw_blue[59/255,145/255,207/255]
mw_navy[0/255,32/255,96/255]
mw_purple[114/255,51/255,161/255]
mw_grey[116/255,116/255,116/255]

It is often useful to put these custom colours in a script so you can call this script before plotting to get your desired colour choice.

plot(t,v,'Color',[255/255,192/255,0/255])

Now we can use the Variable Name in place of the RGB row vector:

customcolours

plot(t,v,'Color',mw_orange)

Because we are using a Variable Name opposed to a Character there is no '.

## LineWidth

As these input arguments come in pairs. It doesn’t matter if we specify the Color first or the LineWidth first:

plot(t,v,'Color',[255/255,192/255,0/255],'LineWidth',5)

plot(t,v,'LineWidth',5,'Color',[255/255,192/255,0/255])

It can also be helpful using ... to split the code over multiple lines. That way you can easily track the pairs:

plot(t,v,...
'Color',[255/255,192/255,0/255],...
'LineWidth',5
)

The lines of code above give:

## LineStyle

MATLAB also has a number of different types of lines, by default the line is a solid line however another two input arguments can be input to change these:

$\displaystyle \begin{array}{*{20}{c}} {\text{Line Style}} & {\text{Specifier}} \\ {\text{solid line}} & \text{-} \\ {\text{dashed line}} & {\text{--}} \\ {\text{dotted line}} & \text{:} \\ {\text{dash-dot line}} & {\text{- }\text{.}} \end{array}$

Once again 'LineStyle' and '--' are both strings of characters and must be enclosed in '  '

plot(t,v,...
'Color',[255/255,192/255,0/255],...
'LineStyle','--',...
'LineWidth',5)

LineStyle
'-'
'--'
':'
'-.'

## Marker

Markers can also be added to the line using two more additional input arguments:

plot(t,v,...
'Color',[255/255,192/255,0/255],...
'LineStyle',':',...
'LineWidth',5,...
'Marker','s'
)

Once again 'Marker' and 's' are both strings of characters and must be enclosed in '  '

MarkerStyle
'+'
'o'
'*'
'.'
'x'
's'
'd'
'^'
'<'
'>'
'v'
'p'
'h'

## MarkerSize

The markers above were quite small, they are hard to read, this can be resolved by using another two additional input arguments:

plot(t,v,...
'Color',[255/255,192/255,0/255],...
'LineStyle',':',...
'LineWidth',5,...
'Marker','s',...
'MarkerSize',20
)

Once again 'MarkerSize' is a string and must be enclosed in '  ' and 20 is a scalar.

## MarkerEdge and MarkerFace

The Marker does not need to be the same colour as the line. We can specify both the Edge and the Fill of the marker with another colour using another four input arguments. Once again 'MarkerFace' and 'MarkerEdge' are both strings and must be enclosed in '  '. Each of them has a paired input argument, which uses the same notation as the line color (see above), I will use the [r,g,b] row vector notation.

plot(t,v,...
'Color',[255/255,192/255,0/255],...
'LineStyle',':',...
'LineWidth',5,...
'Marker','s',...
'MarkerSize',20,...
'MarkerEdge',[0/255,176/255,80/255],...
'MarkerFace' ,[112/255,48/255,160/255]
)

## Short Hand

For learning pairing the inputs may help for using the plot function but it is quite useful to use the abbreviated notation for some of the most commonly used inputs.

plot(t,v, 'color','b','LineStyle',':','Marker','s')

plot(t,v,'b:s')

# Axes

For the axes we use the function set with the input argument gca (get current axes):

set(gca)

## Limits

To change the x axis limits the additional input argument 'XLim' in the for of a string  of characters is matched by a 2 by 1 row vector consisting of the upper and lower x limits respectively. Likewise we have the additional input argument 'YLim' matched input argument is a 2 by 1 column vector (if a row vector is input, it will automatically be adjusted) this time consisting of the upper and lower y limits respectively.

set(gca,...,
'XLim',[10,20],...
'YLim',[400,500])

## Scale

We can also use the additional input argument 'XLim'  to set the scale on the x axis using the accompanying input argument 'Lin' for Linear (default) or 'Log' for logarithmic scale. For the y scale we use 'YLim'

set(gca,...,
'XLim',[10,20],...
'YLim',[100,500],...
'XScale','Lin',...
'YScale','Log')

## FontSize

If we wish to change the Font Size on the axes we can use the additional input argument 'FontSize' followed by a scalar input:

set(gca,...,
'XLim',[10,20],...
'YLim',[100,500],...
'XScale','Lin',...
'YScale','Log',...
'FontSize',20
)

## Grid

One can enable the grid by using the function grid with the input argument 'on'
grid('on')

We can have gridlines with minor gridlines using the input argument 'minor'

grid('minor')

One can disable the grid by using the function grid with the input argument 'off'
grid('off')

Because these are commonly used, they also in the form:

grid on

grid minor

grid off

# Axis Labels and Title

There are three functions to label the x-axis, y-axis and the figure title these are:

xlabel('time (s)')

ylabel('speed (m/s)')

title('Rocket Projectile')

Note how the brackets enclosed within the '  ' are treated as text and not as part of the code.

If alternatively one has a cell array with the labels, these can also be input as axis labels:

mytestlabels={'test time (s)','test speed (m/s)','Rocket Projectile'}

xlabel(mytestlabels(1))

ylabel(mytestlabels(2))

title(mytestlabels(3))

# Legend

The graph is missing a legend we can add one using the following function

legend('Rocket1',...
'Location','NorthWest')

legend('Rocket1',...
'Location','SouthEast')

For the location we use the compass points, these are usually within the chart area but we can also specify for them to be outside.

Location
'North'
'NorthEast'
'East'
'SouthEast'
'South'
'SouthWest'
'West'
'NorthWest'
'West'
'Best'
'NorthOutside'
'EastOutside'
'SouthOutside'
'WestOutside'
'BestOutside'

# Hold

So far we have the following code.

figure(1)

plot(t,v,...
'Color',[255/255,192/255,0/255],...
'LineStyle',':',...
'LineWidth',5,...
'Marker','s',...
'MarkerSize',20,...
'MarkerEdge',[0/255,176/255,80/255],...
'MarkerFace' ,[112/255,48/255,160/255]
)

set(gca,...
'XLim',[10,20],...
'YLim',[100,500],...
'FontSize',20
)

xlabel('time (s)')
ylabel('speed (m/s)')
title('Rocket Projectile')

grid('minor')

legend('Rocket1',...
'Location','SouthEast')

This gives us the following:

Now supposing we want to create a plot of v vs t and also a second Rocket that is twice as fast 2*v vs t:

figure(1)

plot(t,v,...
'Color',[255/255,192/255,0/255],...
'LineStyle',':',...
'LineWidth',5,...
'Marker','s',...
'MarkerSize',20,...
'MarkerEdge',[0/255,176/255,80/255],...
'MarkerFace' ,[112/255,48/255,160/255]
)

plot(t,2*v,...
'Color',[192/255,0/255,0/255],...
'LineStyle',':',...
'LineWidth',5,...
'Marker','s',...
'MarkerSize',20,...
'MarkerEdge',[0/255,176/255,80/255],...
'MarkerFace' ,[112/255,48/255,160/255]
)

set(gca,...,
'FontSize',20
)

xlabel('time (s)')
ylabel('speed (m/s)')
title('Rocket Projectile')

grid('minor')

legend('Rocket1','Rocket2'...
'Location','SouthEast')

This code overwrites the data on Figure 1 opposed to adding a second Line Plot.

To add additional lines to the Figure 1 opposed to writing over them we need to use the hold function with the input argument 'on'. We should use the hold function with the input argument 'off' to release after we’ve finished plotting.

figure(1)

plot(t,v,...
'Color',[255/255,192/255,0/255],...
'LineStyle',':',...
'LineWidth',5,...
'Marker','s',...
'MarkerSize',20,...
'MarkerEdge',[0/255,176/255,80/255],...
'MarkerFace' ,[112/255,48/255,160/255]
)

hold('on')
plot(
t,2*v,...
'Color',[192/255,0/255,0/255],...
'LineStyle',':',...
'LineWidth',5,...
'Marker','s',...
'MarkerSize',20,...
'MarkerEdge',[0/255,176/255,80/255],...
'MarkerFace' ,[112/255,48/255,160/255]
)
hold('off')

set(gca,...,
'FontSize',20
)

xlabel('time (s)')
ylabel('speed (m/s)')
title('Rocket Projectile')

grid('minor')

legend('Rocket1','Rocket2'...
'Location','NorthWest')

Because these are commonly used, there is an abbreviated form:

hold on

hold off