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What is a tensor and why is it useful?

I have grabbed "Vectors, Tensors and the Basic Equations of Fluid Mechanics" by Rutherford Aris, but it is not a gentle introduction (some of the notation used is not explained at all!).

A while back when I decided to learn relativity it became clear to me that I needed to learn tensors. So I learned a good deal about them and decided to post the URL to the site to relativity sites and it seemed to help people who were attempting to learn relativity. The URL is located at http://home.comcast.net/~peter.m.brown/math_phy/math_phy.htm

Energy is, exactly, the conserved current related to time invariance of the Lagrangian. Google "Noether's theorem".

That is not a definition of energy, it's an equality.

 

Studiot - Regarding your response to an attempt at defining energy in terms of the Lagrangian. I highly admire your response to that attempt at a definition. I totally missed that circularity part! Way to go! Bravo my good sir!

 

Energy, like several quantities in physics, is one of those things that goes without a definition. I did some research on energy and wrote up th result of what I thought best suited as a good response to the question What is Energy? As Richard Feynman wrote in The Feynman Lectures (see http://home.comcast.net/~peter.m.brown/mech/what_is_energy.htm)

 

It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not tht way. However, there are formulas for calculating some numerical quantity, and we at it all together it gives "28" = always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.

 

The concept of 'energy' is not confined to Physics only. What about informational energy (Shannon's concepts). Also biological energy (in the form of ATP etc), and metaphysical notions of energy. But, I suspect, if one goes deep enough, there will be common grounds for any type of energy, a 'thing' that makes stuff happen

As I recall, ATP is a conserved quantity which is conserved and whose units are that of that of energy, so really it's a physics kind of "energy".

I saw elsewhere in this thread that someone was bashing the use of analogies simply because the analogy was ultimately flawed in one way or another. I'd like to point out that analogies are just that, analoges by definition is just that a flaw between concepts. If there were zero flaws then there wouldn't be an analogy, it would simply be that, i.e. i would be the thing being compared.

 

I mention this for two reasons before, because someone spent a lot of time trying to prove to that the analogy I was using was flawed and therefore my argument was flawed for that reason. I think that this point should be kept in mind when using analogies. It's also important to keep in mind the limits of the analogy.

elfmotat wrote ------------------------

Your conclusion is correct, but your logic is flawed. Newton's inverse square law is a statement about rest mass, not relativistic mass.

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Newton's inverse square law is merely an approximation to the gravitational field. It's Einstein's General Theory of Relativity which is the correct statement for all values of stress, energy and momentum. It is more correct to place relativistic mass in place of rest mass rather than rest mass which seems to be what you're doing. There is an illustrating example from the American Journal of Physics.

 

Measuring the active gravitational mass of a moving object, D.W. Olson and R.C.Guarino, Am. J. Phys. 53(7), July 1985. The abstract reads

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Abstract - If a heavy object with rest mass M moves past you with a velocity comparable to the speed of light, you will be attracted gravitationally towards its path as thought it had an increased mass. If the relativistic increase in active gravitational mass is measured by the transverse (and logintudinal) velocities with which a moving moving mass induces in test particles initially at rest near its path, then we find, with this definition, tha M_rel = gamma[ 1 + (v/c)^2]M. Therefore , in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not gamma*M but is approximately 2*gamma*M.

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Yes, they do. But that's beside the point. It's not what everyone else is discussing in this thread.

This is a forum to discuss relativistic effects so when someone comes in talking about the slowing down of clocks then it is, by definition, and ideal clock they are referring to. All questions regarding slowing down come from relativity and not good clocks gone bad,

 

Tell ya what. I'll aswer the questions which address the slowing down of good clocks and you answer the questions about broken/poorly synchronized clocks. In any case my response was correct.

 

 

Okay. Enough of this. At this point Suxamethonium wants to know more about bad cocks or he wants to know about ideal clocks. Anything other that the laer I'm not interested.

Ideal clocks behave as ideal clocks.

"You're kidding!? Wow! I never knew that.

I think it was clear that timo wasn't limiting the discussion to ideal clocks by the time "all physical processes slow down" came up since ideal clocks exclude most physical processes. And yes, your response came after timo's and my posts. Causality and all that. But since you responded, you might consider that not changing the context of the discussion would be a good thing.

Ideal clocks come up quite often. Physicists quite often use ideal clocks when working out theortetical problems.

When you assume that all clocks are ideal, then all conclusions you reach are valid for ideal clocks. In case I wasn't clear before: I am not discussing ideal clocks. Thus, your point is moot.

 

I was very clear about the fact that I was talking about ideal clocks when I responded to timo. That you weren’t discussing ideal clocks is the real moot point here since my response to timo was in post 4 and not earlier. And I still don’t see where the tautology that you spoke of was. Please point it out for me. Thanks.

Aren't tautologies grand?

If you believe there was a tautology in my post then why don't you simply post an arguement proving it?

As showed in the other thread Einstein did many mistakes, ...

And I said that everyone makes mistakes so the fact that Einstein made mistakes merely makes him human and nothing else and as such is insufficient reason for you to counter Einstein elsewhere.

...his original thinking does not satisfy current scientific standards of rigour and precision, and his old quotes do not quality as an argument based in 'authority'.

I merely quoted Einstein there for historical context, i.e. to show where the viewpoint "light has matter" started. To say that Einstein's thinking does not satisfy current scientific standards of rigour and precision is unjustified.

 

The standard definition of matter can be found in chemistry and physics textbooks and shows that light is not usually considered a form of matter.

The term "matter" is typically a vauge term which is never really defined in chemistry and physics textbooks.

And in general relativity, matter also follows the spacetime (which has been warped by gravity). One cannot say that matter is affected by gravity whereas light is not, when both are effected and 'bend'.

What do you mean by matter also follows the spacetime ?

No, timos's statement is correct. If I move a pendulum clock from a valley to a hill the clock slows down even though time speeds up, because [math]\sqrt{\frac{g}{L}}[/math] is a much bigger effect than [math]\frac{gh}{c^2}[/math]

I'm way ahead of you swansont. A pendulum clock is not an ideal clock. That's why your counter example fails.

 

Details: Consider using a pendulum clock as a ship's clock in a rocket ship. The rate at which the clock runs will depend on the rate at which the rocket accelerates. That means that a pendulum clock is not an ideal clock.

 

What I'm saying in this thead is not my own personal opinion/definition but the definition as it is found in the relativity literature. For example: see http://home.comcast....s_textbooks.htm for an illustrative list of relativity texts and each one is consistent with what I've said above.

Then you failed to follow through with my question

 

I'm not in the mood for sarcasm or jokes so please just answer my question. Thanks.

My appologies. I just reread my last post in this thread and saw how nasty I might have come across. If that is the case the please acceptmy appologies.

So you're saying that if an sufficient amount non-outspoken extra criteria are applied to B, then A and B become equivalent, and that this then makes "the proper statement is A, not B" incorrect?

Sorry, but I have no idea what that says/asks.

The proper statement is "time slows down", not "clocks slow down".

That is incorrect. The two statements are synonymous. Of course when such statements (or similar statments) are made it is understood that one is refering to an ideal clock where ideal clock is defined as a clock whose operation is unaffected by its acceleration. Also, as stated above by swansont, one ideally one also wishes the ideal clock to be unaffected by temperature, humidity, radiation etc. No use having an idea clock if the cold temperatures of interplanetary space causes the clock to malfunction.

well, we'd come to the same conclusions.

 

we have never directly observed a blackhole, we only know they emit no light because of logical deduction.

Don't forget about the Hawking Radiation comming from the region around a black hole. Also in some cases there are accreation disks around black holes.

I've seen it used before, yes. Off the top of my head, Schutz's Gravity from the Ground Up comes to mind (if you consider it a GR text - it's not exactly rigorous).

http://www.gravityfr.../timecurves.pdf

Hmmm! Interesting. This is one of my favorite texts for gravity. I didn't realize he used this terminology. I guess I should learn the symantics of this terminology regardless of whether I think it's logical or not.

 

Thanks for the reference.

It is not right that only matter is affected by gravity. Light is also affected. For instance light bends around Sun due to gravity.

 

Neither "light is made up of matter".

Physicists understand light to be a form of matter. This viewpoint started with Eintein when he wrote his papers on relativity. For example; in Einstein's 1916 paper which is a review on gravity. In section 14 Einstein writes

We make a distinction hereafter between "gravitational field" and "matter" in this way, that we denote everything but the gravitational field as "matter". Our use of the word includes not only matter in the ordinary sense but the electromagnetic field itself.

I had a rather new undergraduate physics text around here somewhere which defined the term in that way. I can't find it right now. But I know of no valid reason to use the term otherwise. In fact it makes sense to think of the electric field and magnetic field in this way too.

 

Neither "light is made up of matter".

On what basis do you make this assertion?

Of course it wasn't I joke, why wouldn't I write on books and then throw them in the fire for no reason? It's so much fun.

 

Then you failed to follow through with my question

then I don't understand your response. Please rephrase.

I'm not in the mood for sarcasm or jokes so please just answer my question. Thanks.

This seems like more of a problem of language than physics.

I disagree. The purpose of my post was to try to curb the usage of that phrase since it's patently incorrect.

"Curved time" usually refers to when |g₀₀|≠1 in a diagonal metric (though this doesn't necessarily imply R^a_bcd=0).

Thank you but I understand quite well what people mean when they use the phrase "curved spacetime". Otherwise I wouldn't comment on it.

 

This seems like more of a problem of language than physics. "Curved time" usually refers to when |g₀₀|≠1 in a diagonal metric (though this doesn't necessarily imply R^a_bcd=0).

Do you know of any GR text which uses the term "curved time" or "time is curved when" or any variant thereof?

Like I wrote the word on a book then threw it in a fire...no, it was a mistake, suppose to say "heaRing"

If that was a joke then I don't get it. If it wasn't a joke then I don't understand your response. Please rephrase.

The mass and energy of the Sun, for example, warps both time and space in its vicinity.

The phrase "time is warped" or anything similar to it is incorect. A manifold must have more than one dimension for it to be warped.

 

The physics literature has a lot of idosynchraties in it on points like that so many years ago I got tired of repeating myself so I wrote a paper on it. It is at http://xxx.lanl.gov/abs/physics/0204044 for those who are so inclined to learn about the subject of spacetime curvature in general relativity and its relationship to gravity.

Don't know, I just remember heating about it in some abstract physics.

I don't understand. What do you mean when you say heating about it in some abstract physics.

I find this hard to understand, what is the space curving into?

How can space be curved and yet appear flat?

Read http://www.eftaylor.com/pub/chapter2.pdf and you'll know what spacetime curvature is.

It is evident for me that you quoted Einstein as authority in your first post, ..

It is obvious that I'm more qualified to interpet the reasons I post something than you are. So let's just leave it at that.

I believe the argument is that it's not a force because free-fall in a uniform field is an inertial frame.

Actualy that arguement holds for all spacetimes, not just those associated with uniform spacetimes/uniform gravitational fields. When people say that "in GR, gravity is not a force" they mean that the gravitational force is an inertial force. That means that, for a body in free-fall in a gravitational field, the 4-force on that particle is zero even though the gravitational force, a 3-force, on it is zero. They don't say this for non-inertial forces because those have a non-zero 4-force associated with them.

 

Sorry, but Einstein did many mistakes [1] …

 

 

Look back over my posts in this thread and you'll notice that I never said that Einstein was flawless. Nobody who was ever a physicist was ever flawless so I don't know what that comment had to do with anything. In fact I have a book on the subject which a friend on of mine wrote. The book is called Einstein's Mistakes by Hans C. Ohanian.

 

and your appeal to authority (your quote does not give any technical argument) is not enough.

 

 

I disagree. It is obviously is a "technical" argument since the very language in which it is written is soaked with mathematical physics. The purpose of the post was not to appeal to authority but to concern ourselves with the architect of general relativity, i.e. the theory we're talking about.

 

 

 

I have read Einstein's original works on relativity, including his textbook and this could not be used today even for an introductory course in relativity!

 

 

What does Einstein's books and scientific articles have to do with this thread? Those works either define the subject matter, teach the subject matter or both. I posted what I did, not because Einstein was teaching us something, but because Einstein was the one who created GR.

 

…this could not be used today even for an introductory course in relativity!

 

For a couple of his works I couldn't disagree more. For other works a text with more up-to-date notation is better. As far as it actually being used I also disagree. In the lat 90's I happened to be talking to the professor at Harvard who was teaching General Relativity and the text he used was The Meaning of Relativity - Fifth Edition by Albert Einstein, Princeton University Press

 

As far as some other parpers he wrote, they are listed under the Readings in General Relativity section of Exploring Black Holes by Talor and wheeler. This text is used in MITs general relativity course. The name of their GR course is Exploring General Relativity

 

I'll stop here since I would have similar objections to the rest of your responses and similar arguments would hold – and I'm too lazy to continue. Lol!