No doubt, tensors are among the most difficult objects in mathematics.

However, having their properties and rules illustrated and fully explained to you by an outstanding teacher and great genius like Albert Einstein can help learners overcome most of these difficulties.

So now, please, concentrate and let Einstein enlighten you:



5. Contravariant and Covariant Four-vectors


Contravariant Four-vectors--- The linear element is defined by four "components" ds_ν, for which the law of transformation is expressed by the equation







The dx'_σ are expressed as linear and homogeneous functions of the dx'_ν. Hence we may look upon these co-ordinate differentials as the components of a "tensor" of the particular kind which we call a contravariant four-vector. Any thing which is defined relatively to the system of co-ordinates by four quantities A^ν, and which is transformed by the same law






we also call a contravariant four-vector. From (5a) it follows as once that the sums A^σ &plusminus; B^σ are such. Corresponding relations hold for all "tensors" subsequently to be introduced.
(Rule for the additon and substraction of tensors)
Contravariant four-vectors--- We call four quantities A_ν the components of a covariant four-vector, if for any arbitrary choice of the contravariant four-vector B^ν







The law of transformation of a covariant four-vector follows from this definition. For if we replace B^ν on the right-hand side of the equation







by the expression resulting from the iversion of (5a),








we obtain






Since this equation is true for arbitrary values of the B'^ν, it follows that the law of transformation is







Note on a Simplified Way of Writing the Expressions---
A glance at the equations of this paragraph shows that there is always a summation with respect to the indices which occur twice under a sign of summation (e.g. the index ν in (5)), and only with respect to indices which occur twice. It is therefore possible, without loss of clearness, to omit the sign of summation. In its place we introduce the convention:---
If an index occurs twice in one term of an expression, it is always to be summed unless the contrary is expressly stated. The difference between covariant and contravariant four-vectors lies in the law of transformation ((7) or (5) respectively). Both forms are tensors in the sense of the general remark above. Therein lies their importance. Following Ricci and Levi-Civita, we denote the contravariant character by placing the index above, the covariant by placing it below.

You have demonstrated how perfectly abstract you and your daddy can be.
Please draw a picture of four mutually perpendicular lines, all meeting at one point. Better yet, make a model of it and take pictures of various views of it.

How about putting some time in a bottle and sending it to me. Hey,if you had some additional talent, maybe you could write a song about it.

I provide the pictures; and you provide the bottle of time; OK, Curt?
http://books.google.ca/books?id=WAW-4nd-OeIC&pg=PA137&lpg=PA137&dq=drawing+tensors+of+general+relativity&source=web&ots=fq6fUBRCwx&sig=EE94gyXSTpoyT2KV72c11jAkfbA&hl=en&sa=X&oi=book_result&resnum=4&ct=result

SSDD Stan, you are using a book about relativity to validate relativity. Is is no different than using the bible to prove God! Nice to see ya back though!

bob s

As far as teaching of tensors is concerned, Einstein in the above passage has done a poor job. In all likelihood, the only useful knowledge the uninitiated can take home from it is that the indices of the contravariant tensor are up and the indices of the covariant tensor are down!

It's a frequent and common observation that the more the physical theory is false and unrealistic, the more its math is difficult and useless. Einstein's General Relativity is a case in point.

However, there is nonetheless a silver lining for Einstein in it. The abstruseness of its equations make average folks suspect there much more hidden wisdom in his theory than actually is. And as a bonus, this clear abstruseness scares many potential examiners away. We have to remember that Isaac Newton once said that he made his Principia so difficult to follow and enjoy in order to get rid of meddlers like Leibniz and Robert Hooke; ”Principia presented a stumbling block: its extremely condensed mathematical form made it difficult for even the most acute minds to follow”: http://www.answers.com/topic/sir-isaac-newton

Einstein certainly has taken this exercise in obscurity and vagueness to the extreme; Am I right?

If you don't mind me saying so, I think you are right. The idea that understanding the real world can only be done in mathematics is silly.

You explained this to me once before, but tell me again; why is putting time on a fourth coordinate (that is self-contradictory) superior to figuring the time delay along with each distance in 3 dimensions?

Thanks Curt...

The early problems with posting seem to go away now, and all the Network54 servers appear to be up and running. However, the GSJ server is still down.

Anyway, mathematics, in my view, works fine in physics only when it's based upon and derived from the physical phenomena themselves rather than importing it and imposing it upon those phenomena.

So you agree that quantum theory is overcomplicated and disconnected from reality then?

It is mathematics masquerading as an explanation, yet a quick peek under the costume will not make you wiser.

Believe it or not, if you actually learn a tiny little bit about calculus, trig, non-euclidean geometry... the mathematics in SR and GR isn't that complex.

Compared to high school algebra maybe, but not when you delve into theories like M-Theory, or even basic Schrodinger equation extrapolations.

I'm not a natural mathematician, yet with a little study, I was able to see the implications of the Einstein Field Equations. I still cannot produce a satisfying image of reality from quantum mechanic equations, and studying them makes me long for the relative simplicity of the EFE's.

Anyone familiar with this field of research would agree that the degree of difficulty in solving the math is proportional to the generality and the abstractness of the subject matter and has nothing to do with scaring away people.

Furthermore, Einstein's 1916 paper is on physics not mathematics; and a working knowledge of tensors and vectors is assumed on the part of potential readers. And in any case, the math of tensors is complete and available to anybody wants to master it; and it is not that difficult.

Thereupon, any accusation of obscurantism in this regard is necessarily baseless and ludicrous. Since there is nothing scientists love more than to enlighten and communicate their findings with the utmost clarity and the most straightforward statements and calculations.

Three coordinates in space, and three coordinates in time: each space coordinate has a covariant time coordinate. The time coordinate is the delay caused by the finite speed of light.

So you want to move from 4 coordinates to 6?

This is simpler?

Btw, the time delay isn't caused by the speed of light, rather it is a result of being embedded in spacetime.

Just as the constancy of the measured speed of light is a result of our embedding in spacetime.

The constancy of the speed of light is only true if measured within the Earth's electrohydrodynamic sheath. The experiment has not been done, because it would upset the Einsteinian Religion.

It's the primary cause of the Pioneer anomaly.

Max™ said: "Btw, the time delay isn't caused by the speed of light, rather it is a result of being embedded in spacetime."

Curt says: Shave with Ocam's razor, Max™! You are the one who keeps harping on "simple."

Six vectors is really only one concept: Distance and time delay are the same thing, w.r.t. relativity. Does away with the Riemann Eigen vector tense past effort.

There was nothing simple about that post.

The speed of light appears constant, not because of anything about the beam of light itself, but because as you move faster through space, you move slower through time. This affects your measurements in such a way that you will always claim a beam of light travels at light speed away from you.

Max™ said: "This affects your measurements in such a way that you will always claim a beam of light travels at light speed away from you."

Curt says: So how does one actually "register" a beam of light moving away from oneself? I suppose if one's eyes were really far apart, and facing each other, and one's mind were really quick; this could be done. I don't doubt that Max™ qualifies for the eyes requirement, but the quick mind; nope, have to find someone else.

I'm sure if you attempted to do so, even someone as clueless as you about relativity could think up some ways to measure the velocity of a beam of light.

Re: Max, Contravariant & Covariant Four-vectors October 25 2008, 7:14 AM

Max,
"The speed of light appears constant, not because of anything about the beam of light itself, but because as you move faster through space, you move slower through time. This affects your measurements in such a way that you will always claim a beam of light travels at light speed away from you."

Max, I have to admire the fact that you can make such a statement on a public forum. I am going to save it for future reference. In fact. I am going to go back over some of your more memorable posts. I detect a book in the making.

I may be clueless, But I know that unless light travels in circles, or is bounced off a mirror, you won't "register" it. Oh yeh, fiber optics. But then, it is known that light travels at different speeds within the cable, defeating the purpose. Oh, well.

I forgot; In Max™'s twenty year old universe, you can see light traveling away from you.

Here, let me help you understand something.


Let's imagine I'm moving towards a polarized photon detector. When this detector picks up a photon with a specific polarization, it makes a record of the time it detected it, and my location.

It also signals to me when it makes a detection, containing the time/location information, with a uniquely polarized photon.

This detector is 1 light-second (300,000 kilometers) away when I send my signal to it.


It's more complicated than the simple mirror experiments which any basic student of relativity should have understood, but I'm trying to make it more obvious what is happening.


I have a polarized photon emitter, and as I move towards this detector, I take a measurement of the time it takes for me to send and receive signals from it.


Let's say that I was moving at 150,000 kilometers per second, I measure the time it takes for the photon to reach a detector and a confirmation photon to return, and from that I could calculate the speed of light while in motion.


Let's consider the reference frame of the detector, it gets a photon sent when I was 300,000 kilometers away. I am 150,000 kilometers away when it detects it a second later. It sends a signal back which I receive roughly 1/3rd of a second later, (during which I crossed 50,000 km, and the light crossed 100,000 km).

It measures the duration of this exchange as 1 and 1/3rd seconds.


In my reference frame, according to Relativity, I'd claim the light beam was moving at 300,000 km/s relative to myself, towards the detector (not 150,000 km/s), and 300,000 km/s relative to myself on the turn trip, (not 450,000 km/s).

Thanks to time dilation, I claim to have traveled 50,000 km when the signal reached the detector, and 50,000 km when the return signal arrived. Measuring the duration of the exchange as 2/3rd's of a second.

When I check my measurements against the detector's, I am confronted with a discrepancy. The detector shows I traveled 200,000 km between sending and receiving, at 150,000 km/s. My clocks, and own observations were influenced by time dilation at that velocity. I thus gauged the distance I traveled per second differently, as my second was dilated relative to that of an observer at rest.

All these measurements: did a surveyor go out and previously establish them?

How does the "at rest" detector know he's at rest? He could just as well be the one "moving." Then he too would find the discrepancy.

It's relative motion, Max™. That's why it's called "Relativity."

It's like I've explained too many times to you: Changing distance causes a continuous change in the time delay. It's a mutual thing between the two "signaling" and "registering" objects, whose distance can be doing one of three things: increasing, decreasing, or not changing at all.

One is not "standing still," the other is not "moving," Max™. The distance between them is changing.

If one object was receding, and then changed direction and began approaching, Then, Max™, then the two objects would have different stories about what happened.

However, in your scenario, both you and your counterpart: "1 light-second (300,000 kilometers) away," will have identical stories, provided he could know the same starting information as you.

For the sake of argument, this entire exchange occurs in a huge interstellar hallway marked off at kilometer intervals. The detector has already been determined to be at rest and not suffering marked time dilation.

So, a surveyor has marked off a hallway from the center of the universe at 1 kilometer intervals. We assume he has the secret map of the universe to know exactly where this is.

So, the next part: "The detector has already been determined to be at rest and not suffering marked time dilation."

How long is the nap? When does the detector wake up? What is the cure for time dilation? Is it a virus? is it a bacteria? Is it genetically passed to generations? What is the mark? A birth defect?

I think your universe needs a good dose of mineral oil. Mom said that this is sometimes better that a good enema.

Your post shows a complete lack of understanding of the simple principles upon which science, not just relativity, rests.

It is hard to talk with you when you know nothing.

So why do you keep talking? Does it hurt so good?

It's hard to talk with you, since you already know everything. Especially when everybody else already knows that you don't.

I dunno, I seem to have kept several of you engaged for well over a month now. Apparently I am fairly easy to talk too.

Now, I unfortunately am not so easy to understand all the time, and I am sorry about that. I try to avoid losing people, but some of these concepts appear completely different in my head than traditional views would suggest.

I'm trying to describe algebra to you, after having learned about the deep logical structure of mathematics, while you're still learning your times tables apparently. Something which seems self-evident to me, may have a completely different and very confusing meaning to you.

Time, for example.

Max™ said: Btw, the time delay isn't caused by the speed of light, rather it is a result of being embedded in spacetime.

Curt says: Yeh, right Max™! So, the finite speed of light doesn't cause time delay between the emission and "registration" of events??? How absurd. Your desire is to make a simple obvious, proven concept obscure. So much for Ocam's razor. You just dulled the blade.

Curt, you poor confused thing.

Let me completely destroy your world view, ok?


Let's say you measure off a period of time while moving a certain distance, and while I am at rest, I measure the period of time I observed for your trip.

Then we both come to a rest, and compare our measurements.



WHAT DO YOU THINK WE WILL FIND?


Hint: the time delay between me observing your motion, and our actual measurements of the motion, doesn't actually matter.

Max, A broken record is not a new achievement, It is an old timey grooved disc with variations in the groove which can be transformed into sound. When one of them is scratched or broken, the transducer just sits in one cycle of revolution, reciting the same thing over and over; just like you, mindlessly repeating you silliness,

I take it thinking is too much for you, sorry bout that kiddo.

Figured you were more mentally matured by now.

Eat the dust, Max.

Just to clarify it for you, since you're incapable of complex thought apparently.


If you were in motion and measured the duration of your trip, and I was at rest and measured the duration I witnessed for your trip.


Then you came back over to me, and we compared them, assuming your trip was rapid enough/our clocks were accurate enough, we would find that you had recorded less time for your trip.


In fact, we could set up experiments showing you had experienced less time during your trip.



This is why GPS clocks have to be adjusted to compensate for ground clocks running slower, it is as though we are constantly accelerating relative to them.



Got anything to say in response, or are you reduced to unweighted denials, and baseless claims?

Max, dream on, boy.

So you have no valid arguments I take it, you sad stupid little child.

I'm so disappointed, in you and bob, I clearly attributed more understanding to the two of you than either possessed.

Max said: "So you have no valid arguments I take it, you sad stupid little child. I'm so disappointed, in you and bob, I clearly attributed more understanding to the two of you than either possessed."

Curt says: Max, "you sad stupid little child," you have to use valid logic to show a mistake in someone else's logic. You haven't done anything of the sort. You use invective, innuendo, and ad hominem attacks to make you think you have accomplished something. Reciting your Origami Universe theory time after time is quite boring. Your problem is that you "attributed more understanding to" the two of us than you posses, yourself.

You need a class in critical thinking and logic. A little philosophy would help too.

So, Max, You've decided to study Philosophy. Good. Who are you working on now?

You're mixing up the conversations.

That is a continuation from another thread, and I didn't say I was studying, I said I have studied.

No Max! what you said was "I do study philosophy actually," on November 28, 2008, at 4:22 PM. You can't even keep straight what you say.

Technically I study Relativistic Concepts as well, but I'm not actively pursuing those studies.

The statement "I study..." does not necessarily imply a current course of study.

As I said, and will readily clarify, I've studied philosophy, I study many things, I have not finished my studies in philosophical concepts, as they are still evolving, and so I do not claim that it is a finished area of knowledge.

As any person possessed of any wisdom would recognize, only a fool claims to have nothing else to learn.

Only a fool like Max claims to know it all, and proclaim himself judge of what the forum shall discuss!!!

If I was currently actively pursuing the study of philosophy, I would have stated: "I am studying philosophy, actually."

Just to be even more clear, I do so wish to avoid confusing the already befuddled posters here. Much prefer to assist you all in relieving yourselves of such stumbling blocks.

iggymax said: "Just to be even more clear, I do so wish to avoid confusing the already befuddled posters here. Much prefer to assist you all in relieving yourselves of such stumbling blocks."

Yeh, right, iggymax, You are the all-time befuddled poster and stumbling block. The rest of us simply go around you and leave you in the dust. That way you can tell yourself how great you are and not be bothered. We are considerate.

General relativity is an exact theory of gravitation as exact as it can be.
It has been used right from the beginning to compute the perihelion advance of the planet Mercury, the minute gravitational red shift in light from the sun, and the bending of starlight by the sun. And it's employed today in making precise mathematical modeling and accurate predictions about neutron stars, quasars, and black holes.

Very briefly, general relativity is an excellent theory and excellent tool for doing physics; and the entire field of astrophysics could never exist or be done as brilliantly as it's now without it.

So, please please please try to rid yourself of those lopsided opinions and love it!

Einstein is using the futile math of his General Relativity to hide the absurdities of his General Relativity. There can be no doubt about that.


Moreover, this tensor math is for the most part math in name only. It's, in the final analysis, a mere complex notation and empty rules with little or no prospect at all of reaching exact solutions or making meaningful calculations.


It follows, therefore, that anyone, who believes this sort of esoteric notation has some deep meaning in it, is a mystic at heart and not a scientist by any measure.


Einstein and his followers have repeatedly made the claim that General Relativity is an exact theory of gravitation. But even a simple problem, like the approximate (not the precise) combined trajectory of the earth and the moon around the sun, cannot be worked out or computed in any meaningful way on the basis of their theory of General Relativity.


Thus, for all practical purposes, the theory of General is good for nothing and useless.

AAF: But even a simple problem, like the approximate (not the precise) combined trajectory of the earth and the moon around the sun, cannot be worked out or computed in any meaningful way on the basis of their theory of General Relativity.


Ted: As always, you are wrong. Just because you are too ignorant to learn even the simplest math doesn't make relativity wrong. It simply makes you stooopid :lol:

AAF: Even a simple problem, like the approximate (not the precise) combined trajectory of the earth and the moon around the sun, cannot be worked out or computed in any meaningful way on the basis of their theory of General Relativity.


Ted: As always, you are wrong. Just because you are too ignorant to learn even the simplest math doesn't make relativity wrong. It simply makes you stooopid :lol:


AAF: 'LOL' Idiot Ted---->>>>>>.....>>>
Can you use this dumb math of GTR to calculate the combined trajectory of the earth-moon system?
If you do and show us the calculations, then I won't call you an 'idiot' anymore; I'll call you 'lousy moron' only; deal? Otherwise, you're just a 'loud-mouth 'Schnook'!

AAF, you simpleton-moron, for someone who thinks that tensorial calculus is just a notation (you didn't manage to get past the introduction paragraph) and that light speed coming from accelerated ions is 2c, you are pretty funny. As a clown, Bozo

Idiot Ted---->>>>>>
We can't get past the introduction, because this math editor doesn't work on this forum anymore:
http://www.codecogs.com/components/equationeditor/equationeditor.php

Iggy Liggety, you are the simpleton-moron, clown, bozo. You can't think outside your buns.