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Thread: SPLIT : General Relativity Primer Discussion

  1. #1 SPLIT : General Relativity Primer Discussion 
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    Quote Originally Posted by Markus Hanke View Post
    I have recently been approached by members of this forum with the question of whether I would be willing and able to put together a thread on the physical and mathematical principles of Einstein's General Relativity, in the "ongoing presentation" style of Guitarist's excellent threads in the maths section. I have decided to try and give this a go, however, I need all readers, whether casual or not, to be aware of the following :

    1. General Relativity ( GR ) is a vast and difficult area, so all I can do here is present the basic building blocks
    2. This thread is not a substitute for an in-depth study of a relevant textbook. I recommend "A first course in GR" by Schutz, as well as "Gravitation" by Thorne/Wheeler/Misner.
    3. This thread is meant as a "primer" on GR; as such I will favour clarity and physical content over mathematical rigour and proofs, which, let's be honest, few people on this forum would be able to follow anyway. Therefore please do expect a lot of ad-hoc derivations and handwaving, and don't try to eat me over lack of rigour on small details. Please do point out obvious mistakes though !
    4. I will try to keep things as simple and laymen-friendly as possible, however, please do understand that GR is an advanced topic, so a certain level of maths cannot be avoided. I will assume that the reader has at least basic knowledge of multi-variable and vector field calculus, without which trying to tackle GR makes little sense. Genuine questions will be welcome throughout the presentation.
    5. Finally, this is a GR primer, it is not a thread for the discussion of anti-relativity ideas ! If you have a beef with the principles and validity of GR, then please open your own thread on it and don't spam this presentation ! If I get inundated with "relativity is wrong" cranks on here, I will have no choice but to simply discontinue the presentation.
    This is an excellent idea.
    Markus,

    Is there any way to delete cranks posts? This way the thread would be kept clean, devoid of their "contributions".


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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by xyzt View Post

    This is an excellent idea.
    Markus,

    Is there any way to delete cranks posts? This way the thread would be kept clean, devoid of their "contributions".
    xyzt, unfortunately I am not a moderator on this forum, so I do not have that power. All we can do is use the "Report" function on the bottom left of each post to see if the moderators can move "crank" posts away from here. Feel free to make use of this as appropriate.


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    Quote Originally Posted by Markus Hanke View Post

    Or another example :


    Apologize for Nitpick:

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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by xyzt View Post
    Quote Originally Posted by Markus Hanke View Post

    Or another example :


    Apologize for Nitpick:

    A silly typo on my side - well spotted, and thanks for pointing it out
    Corrected.
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    Quote Originally Posted by Markus Hanke View Post

    For a contravariant vector ( upper indices ! ) to be independent of the coordinate basis, its elements must contra-vary with our change in coordinates. For example, imagine a position vector which is 1000m long; we are now performing a ( rather trivial ) transformation of our coordinate system from meters to kilometres, i.e. the basis vectors become longer in our new system. To compensate, the original vector must become shorter ( magnitude goes from 1000 to 1 ) to compensate - it contra-varies in the opposite sense to represent the same physics. Examples of contravariant vectors would be anything to do with positions, and its derivatives like velocity, acceleration etc.

    For a covariant vector ( lower indices ! ), the situation is reversed - for it to retain its physical meaning under a coordinate transformation, it must co-vary ( vary in the same sense ) with the basis vectors. This is generally the case if we are dealing with gradients, and such vectors would have dimensions inverse to the ones listed above, so for example "per meter".
    A simple (simplistic) rule to distinguish the two types is to associate tangents to the manifold with contra-variant vectors and normals to co-variant vectors. Since tangents and normals don't mix, neither do contra- and co- variant vectors.
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    Yeah awesome explaination by the way
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    Quote Originally Posted by Markus Hanke View Post

    (scalar) = (tensor) x (vector) x (vector)
    .

    Lastly I will show you how to raise and lower indices on tensors. The basic idea is very simply to sum the index in question with the opposite index of the metric tensor defined on our manifold; i.e. to "lower" an upper index, you sum it up with a corresponding lower index of the metric tensor, thereby eliminating it and leaving only a lower index, and vice versa :



    and



    Likewise for indices on rank-2 tensors :


    Again, a simple (simplistic) way to think about it is to refer to 3D space:

    -the product between a tensor (represented by a 3x3 matrix) and a vector is a vector
    -the further product between the vector resulting from the above operation and another vector is a scalar (think of the dot product of two vectors in 3D)

    At each step, you can see how the rank is reduced :

    -from rank-2 tensor (matrix) to rank-1 tensor (vector)
    -from rank-1 tensor (vector) to rank-0 tensor (the scalar resulting from the dot product of the two vectors)
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    Sorry Markus, I cannot stop myself.

    First this, though..... The fact I don't like your characterization of the trace of a tensor you can put down to my notational fastidiousness.

    But this does not seem quite right
    Quote Originally Posted by Markus Hanke View Post

    Consider now a general 4-vector A on our 4-dimensional manifold - locally, since by above definition of a manifold we can approximate a local neighbourhood by ordinary Euclidean space, any such vector can be decomposed in terms of the basis vectors of our coordinate system by multiplying each one with a scalar :

    Note that the scalars you use are caled the "components" of your tensor. Now in the (ghastly) notation you learn from your physics texts you will find any tensor written ONLY in terms of its components - so that refers to the components - the scalar coefficients - on a set of basis vectors. That is the above should be written as
    So your assertion seems not to be quite correct - if you insist on referring tensors (in this case vectors) by their components, then you have indices inconsistent!

    Now, it is important to reiterate that this is a local decomposition, so with a BIG wave of the hand, and totally without rigorous proof, I now state that the set of basis vectors can be written as

    Ok, you may say notation is arbitrary, but we are talking about some sort of tangent space to a differentiable manifold, right

    I assume by your notation you intend the vector , what is rather outdatedly called a "contravariant vector". But the basis for the space of all such vectors at each point is referred to as the directional derivatives at that point - differential operators to be exact - as the set whereby any may be written in your notation as assuming implied summation

    The set is in fact basis for the "companion" space , the space of all linear functions , whose elements - co-tangent vectors - are, by your component notation, written
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    Quote Originally Posted by Guitarist View Post
    Sorry Markus
    Not at all, Guitarist. All constructive contributions are more than welcome ! I am making all of this up as I go from my own understanding of things, so I do depend on other readers to point out errors to me.

    That is the above should be written as
    Noted, agreed, and changed in my post.

    Ok, you may say notation is arbitrary, but we are talking about some sort of tangent space to a differentiable manifold, right
    Basically yes. I am trying to justify why, in the definition of the squared line element , the "dx" terms can be considered vectors. Many readers not well versed in maths simply will not be able to realize this; it was a major stumbling block for me when I first started to learn about GR. It is all good and well to think of tensors as "functions" which have vectors and covectors as input, but if it is not clear why "dx" terms can be regarded as such, then the aforementioned expression for the line element makes little to no sense.
    I have amended the passage in my post like so - I hope this makes more sense :





    How would you, from a mathematician's point of view, have explained that ? The line element, once understood, is actually a very intuitive concept, but I find that trying to explain it to laypeople without getting lost in mathematical abstractions is really quite difficult. Personally, I simply think of the "dx" terms as infinitesimal vectors, and the whole expression then makes sense to me.

    P.S. I might need a little input on the next topic, connections and covariant derivative. It is very difficult to explain this to non-mathematicians, and I am not clear myself on some of the details.
    Last edited by Markus Hanke; May 6th, 2013 at 02:36 AM.
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    Brassica oleracea Strange's Avatar
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    Thanks for this Markus. I am working through it slowly and (I think) making sense of it.

    I have just got to this:

    Quote Originally Posted by Markus Hanke View Post
    Consider now a general 4-vector A on our 4-dimensional manifold - locally, since by above definition of a manifold we can approximate a local neighbourhood by ordinary Euclidean space, any such vector can be decomposed in terms of the basis vectors of our coordinate system by multiplying each one with a scalar :

    Is this "basis vector" of which you speak, the same thing as the coordinates () mentioned previously?
    Without wishing to overstate my case, everything in the observable universe definitely has its origins in Northamptonshire -- Alan Moore
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    I love relativity and find it most interesting topic in Physics.
    I am agree with it except two interpretations which I think logically false
    logically true true facts are
    1. At a particular point of time every particle at same time share the same "present" in space
    2. No particle can move into past even if it travels at speed of light
    I believe both are logically true
    "No law of Physics is surprising & can not beat commonsense until it does not give enough explanation logically or I did not understand it rightly or simply it is wrong "
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    Quote Originally Posted by Strange View Post
    Thanks for this Markus. I am working through it slowly and (I think) making sense of it.

    I have just got to this:

    Quote Originally Posted by Markus Hanke View Post
    Consider now a general 4-vector A on our 4-dimensional manifold - locally, since by above definition of a manifold we can approximate a local neighbourhood by ordinary Euclidean space, any such vector can be decomposed in terms of the basis vectors of our coordinate system by multiplying each one with a scalar :

    Is this "basis vector" of which you speak, the same thing as the coordinates () mentioned previously?
    No, basis vectors and coordinates are not the same thing. is akin to
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    Quote Originally Posted by Markus Hanke View Post
    6. CONNECTIONS AND COVARIANT DERIVATIVES





    The single pipe before the index denotes the ordinary partial derivative with respect to the k-th coordinate ( I will talk about this particular notation a bit later ).

    Now let us perform that same thought experiment on a different kind of manifold, the 2-dimensional surface of a ( 3-dimensional ) sphere. Let us pick an arbitrary point A on the sphere's surface - the tangent space at that point would be a flat plane, which is why it is unsurprisingly called the tangent plane. Now pick another point B on the same sphere's surface, and again visualize the tangent plane at that new point. Finally, compare those two tangent planes, and you will notice immediately that they are not parallel; instead they are at an angle relative to each other. What that means is that, if we naively calculate the derivate along the same direction but on different points of the surface, we get a different result ( a vector with the same magnitude but pointing in a different direction ). That is rather awkward, since it makes it a lot harder to define a differential operator for the "take the derivative with respect to the k-th coordinate" which gives comparable results no matter where on the surface we are. The above expression for the ordinary partial derivative is clearly insufficient for this purpose. If we wish to find a differential operator which generalises the notion of "partial derivative", and which is equally valid on all points of the manifold in question, we have to somehow compensate for the change in orientation of our tangent plane as we go from point A to point B. We will denote this new, generalized operation by a double pipe before the index :


    A simplistic way of understanding covariant derivatives is to remember what we know from calculus about total derivatives. The connection coefficients can be understood like the coefficients in the "chain rule" of partial derivatives. Again, this is a very simplistic way, introduced just to help understand the math.
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    Quote Originally Posted by Strange View Post
    Is this "basis vector" of which you speak, the same thing as the coordinates () mentioned previously?
    No. The coordinates are the labels which we give the axis in our coordinate system; for example, in the case of Cartesian coordinates that would be {x,y,z} in 3 dimensions. The basis vectors would be vectors which lie along the coordinate axis; for simplicity we make them, say, one unit long. For the above mentioned Cartesian system the basis vector would then be







    Any vector can then be expressed as a sum of the basis vectors multiplied by a constant, in the manner described in the post.
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    Oh dear! What confusion there is here!

    I blame the evil notation that Markus has chosen to use - on the one hand we have that is a tensor, on the other hand it refers to scalar coefficients on basis vectors.

    I do NOT blame Markus for this, it is standard in physics texts. The fact I don't like it is immaterial: the fact that it leads to confusion is undeniable based on the evidence.

    I cannot see how to recover this thread from here - maybe I should start a companion thread on tensor analysis on manifolds. Dunno, as I have covered all that stuff on this site more than once.

    Anyhoo, just for the record xyzt: do not confuse the total derivative with the absolute differential, which before Einstein came up with principle of general covariance was the name given to what is now usually called the covariant derivative.

    The chain rule CAN be used to extract the covariant derivative, but it is not the only way. On the other hand the chain rule is integral to the definition of the total derivative. I repeat, they are not the same
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    Quote Originally Posted by Guitarist View Post
    I do NOT blame Markus for this, it is standard in physics texts.
    That's a relief
    I am not making all this up, this is really how it all appears in physics textbooks. But in fairness now, I would not be able to derive everying in a mathematically rigorous, and notationally acurate way. Attempting to do so would lead to a textbook of several hundred pages, which is beyond the scope of this forum ( and I don't have the maths knowledge anyway ). The main idea here is merely to get across the basic ideas of the "building blocks" of GR. It is not an easy task to condense this down enough to be presented on a thread like this, so I am trying my best.

    If anyone spots any really serious errors, both in notation and understanding, please do point them out to me. That is crucial !
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    Quote Originally Posted by Guitarist View Post

    Anyhoo, just for the record xyzt: do not confuse the total derivative with the absolute differential, which before Einstein came up with principle of general covariance was the name given to what is now usually called the covariant derivative.

    The chain rule CAN be used to extract the covariant derivative, but it is not the only way. On the other hand the chain rule is integral to the definition of the total derivative. I repeat, they are not the same
    I don't confuse anything and I did not claim the two to be the same, I simply showed an analogy between covariant derivative (a more complex notion) and the more familiar notion from calculus.
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    Quote Originally Posted by Markus Hanke View Post

    This is called the Exterior Schwarzschild Metric, and its form is the simplest possible vacuum solution to the original field equations without cosmological constant. We will discuss its properties in more detail in the next post.
    Markus, could you add the discussion on the Internal solution as well?
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    Quote Originally Posted by xyzt View Post
    Quote Originally Posted by Markus Hanke View Post

    This is called the Exterior Schwarzschild Metric, and its form is the simplest possible vacuum solution to the original field equations without cosmological constant. We will discuss its properties in more detail in the next post.
    Markus, could you add the discussion on the Internal solution as well?
    Yes, I can. In fact it was on my agenda anyway, since that is what we derive the dynamics of collapsing stars from. I will dedicate a separate post to it, but I won't show the explicit derivation of the interior SM from the field equations since it is very lengthy and tedious; I'll just give the initial conditions and present the final solution.

    Anything specific you are after ??
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    Quote Originally Posted by Markus Hanke View Post
    I will dedicate a separate post to it, but I won't show the explicit derivation of the interior SM from the field equations since it is very lengthy and tedious; I'll just give the initial conditions and present the final solution.
    Thank you :-). (and yes, I know the derivation0

    Anything specific you are after ??
    Yes, I am interested mostly as to how the solution applies to experiments conducted inside gravitational bodies, like in tunnels drilled radially.
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  22. #21  
    Moderator Moderator Markus Hanke's Avatar
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    My apologies to everyone who might be following this thread; I have been too busy lately to keep working on this. I still have much to present, so I will come back to it as time permits.
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