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Apashanka Das
In the geodesic equation why is dt/du=λ(u)t ,where t is the tangent vector along the curve
and why Dt[a]/Du=λ(u)dx[a]/du equated same,as given in hobson
and why Dt[a]/Du=λ(u)dx[a]/du equated same,as given in hobson
Apashanka Das said:Mr. Orodruin can you please explain the first elaborately ,its very much needed
Apashanka Das said:please give some precise explanation
Apashanka Das said:I have read Hobson
I am sorry, but it is impossible to decipher what you are actually asking about. Can you reformulate the question in such a way that it becomes clear what you want to know? Please explain what it is in the equations you listed that gives you problems. If you are reading these things, I assume that you are at least relatively familiar with calculus on manifolds?Apashanka Das said:Nothing to tell just a post graduate student at st xaviers college ,the question strikes me that's why I m asking ,I have read Hobson
Since I don't have the book, and some pages (including the one you reference) are not accessible to me via the link, I have to intuit the meaning. I agree with you and state as much for the the interpretation of D. However, several usages of d applied to a vector without a coordinate basis are all consistent (IMO) with this being an abstract absolute derivative, as distinct from d applied to components which is always an ordinary derivative. The discussion from 3.39 to 3.41 seems consistent with my interpretation, but I cannot see pages where certain quantities are formally defined. In any case, I find the notation and ambiguity about what derivative is meant pointlessly confusing. Note also the discussion of covariant derivatives on page 69. To me, this also supports my interpretation that Hobson uses ordinary derivative symbols applied to an abstract vector as equivalent to the covariant (or absolute) derivative. See especially just before 3.32 on this page.strangerep said:Actually, Hobson uses big D for the intrinsic (or "absolute") derivative along a curve. [See eq(3.37) in section 3.14.]
@Apashanka Das : I guess the 1st part of your question is about why the notion of "geodesic" is defined by the requirement that the intrinsic derivative of the tangent vector ##{\mathbf t}## along the curve should be a multiple of itself, i.e., why
$$\frac{d{\mathbf t}}{du} ~=~ \lambda(u) {\mathbf t} ~,$$where ##u## is an arbitrary parameter. Then, since this ordinary derivative is not covariant, Hobson switches to the intrinsic derivative ##D/Du## which involves the connection ##\Gamma##, ending up at eq(3.45).
Aaargh! You are right -- this is clear from p72. (I tried to insert corrections in my earlier post but it was too messy, so I deleted the entire post to avoid confusing anyone else.)PAllen said:Since I don't have the book, and some pages (including the one you reference) are not accessible to me via the link, I have to intuit the meaning. I agree with you and state as much for the the interpretation of D. However, several usages of d applied to a vector without a coordinate basis are all consistent (IMO) with this being an abstract absolute derivative, as distinct from d applied to components which is always an ordinary derivative.
No, what is confusing is the use of ordinary derivative symbols for covariant or absolute derivatives when components are not used. I have literally never seen this before. Your derivation provides some motivation, however all books I've seen use some symbol other than ordinary derivative symbols for abstract coordinate free equations.vanhees71 said:Hm, but why is the (very common) notation of covariant derivatives for components confusing? It's used for decades in differential geometry, general relativity, and gauge-field theory...
But to define ##\mathrm{d} \boldsymbol{V}/\mathrm{d} \lambda## you need additional structure. Just the differentiable manifold is not enough you need a connection. That's why it is more common to denote it ##\mathrm{D} \boldsymbol{V}/\mathrm{d} \lambda##.vanhees71 said:But the point is that if ##\boldsymbol{V}(\lambda)## is a vector and ##\lambda## is a scalar, then ##\mathrm{d} \boldsymbol{V}/\mathrm{d} \lambda## is a vector. So why should you introduce another symbol for such a derivative than the usual one? My motivation was to make clear, why you don't need a special symbol for the vectors themselves and why you need one for the components.
And now, with more thought, I question your whole justification. Independent of any coordinate basis, there is no notion of vector derivative without a connection. Otherwise there is no way to compare or subract vectors at different points, even along a given curve, because the vectors are in different tangent spaces - and the connection is what maps between tangent spaces of nearby points. Given this need for connection to even define a vector derivative, it is confusing to use an ordinary derivative symbol for this.vanhees71 said:But the point is that if ##\boldsymbol{V}(\lambda)## is a vector and ##\lambda## is a scalar, then ##\mathrm{d} \boldsymbol{V}/\mathrm{d} \lambda## is a vector. So why should you introduce another symbol for such a derivative than the usual one? My motivation was to make clear, why you don't need a special symbol for the vectors themselves and why you need one for the components.
That makes me wonder: why then are we allowed to subtract scalars at different points?PAllen said:[...] Independent of any coordinate basis, there is no notion of vector derivative without a connection. Otherwise there is no way to compare or subtract vectors at different points, even along a given curve, because the vectors are in different tangent spaces - and the connection is what maps between tangent spaces of nearby points. [...]
strangerep said:why then are we allowed to subtract scalars at different points?
Well, I was thinking of the case of a scalar-valued function ##f(\lambda)## defined on a curve ##\gamma^i(\lambda)##. Calculating its derivative involves ##f(\lambda+\epsilon) - f(\lambda)##.PeterDonis said:Can you give an example?
Yes, that's similar to what I was thinking. But... there's also more than 1 possible isomorphism between different copies of ##\mathbb R##, unless you invoke some extra criteria. E.g., 0 matches 0, and the mapping is isometric (hence no scale changes allowed).martinbn said:The scalars have values real numbers. We can subtract real numbers. The vectors belong to different vector spaces. They may be isomorphic, but there are many different isomorphisms. How do you pick an isomorphism? How do you subtract ##v\in T_pM## and ##w\in T_qM##? On the other hand ##\alpha\in\mathbb R## and ##\beta\in\mathbb R## can be subtracted.
PeterDonis said:Are we? Can you give an example?
PAllen said:And now, with more thought, I question your whole justification. Independent of any coordinate basis, there is no notion of vector derivative without a connection. Otherwise there is no way to compare or subract vectors at different points, even along a given curve, because the vectors are in different tangent spaces - and the connection is what maps between tangent spaces of nearby points. Given this need for connection to even define a vector derivative, it is confusing to use an ordinary derivative symbol for this.
[edit: I see I cross posted with @martinbn ]
Unambiguous, yes , if defined by the author. Confusing and nonstandard, also yes. Also, my argument was in response to an argument that connections were only relevant in a coordinate basis - which is not correct.stevendaryl said:Okay, but in the context of talking about a smooth manifold with connection (which is the case for GR), the notation [itex]\frac{d\mathbf{V(s)}}{ds}[/itex] is unambiguous.
Why then don't you need a connection when dealing with plain differentiable manifolds without any additional structures? You can go pretty far, introducing tangent and cotangent spaces and introducing differential forms. For this I don't need a connection, let alone a fundamental form (i.e., metric of pseudometric).PAllen said:And now, with more thought, I question your whole justification. Independent of any coordinate basis, there is no notion of vector derivative without a connection. Otherwise there is no way to compare or subract vectors at different points, even along a given curve, because the vectors are in different tangent spaces - and the connection is what maps between tangent spaces of nearby points. Given this need for connection to even define a vector derivative, it is confusing to use an ordinary derivative symbol for this.
[edit: I see I cross posted with @martinbn ]
Why would you?vanhees71 said:Then, why don't I need a connection to define (alternating) differential forms?