Recent content by schwarzschild

If \tilde{p} is a one-form, then does \tilde{p}(\vec{a} - 3\vec{b}) = \tilde{p}(\vec{a}) - 3\tilde{p}(\vec{b})?

In Schutz's treatment of general relativity he defines a one-form as a function which maps a vector to a real number, and then later defines a vector as a linear function that maps one-forms into the reals. So the definitions seem to be circular - is there another way we can define a vector?

Are the following three equivalent? P_{\alpha}A^{\beta}\tilde{\omega}^{\beta}(\vec{e_{\beta}} ) = \sum_{\alpha = 0}^{3}{P_{\alpha}\tilde{\omega}^{\alpha}(\sum_{\beta = 0}^{3}{A^{\beta}\vec{e}_{\beta}) = \sum_{\alpha = 0}^{3}{P_{\alpha}A^{\alpha}

I'm having trouble understanding the following sentence from Schutz's A First Course in General Relativity, so I was hoping someone could explain/expound on it. " p_a \equiv \widetilde{p}( \vec{e_{\alpha} ) } ) Any component with a single lower index is, by convention, the component of a...

Suppose you have an question like: "In the t-x spacetime diagram of O, draw the basis vectors \vec{e}_0 and \vec{e}_1 Draw the corresponding basis vectors of \bar{O} , who moves with speed 0.6 in the positive x direction relative to O. Draw the corresponding basis vectors of...

I'm not sure I understand how: T^{\alpha \mu \lambda}A_{\mu}C_{\lambda}^{\gamma} = D^{\gamma \alpha} "Represents 16 different equations..." My thinking was that \alpha and \gamma each have four possible values \left\{0,...3 \right\} so we have 4 \cdot 4 = 16 different...

Wow! Thanks for pointing that out - the two are confusingly similar in appearance :biggrin:.

I thought that when you used a roman letter such as v that you started at 1 instead of 0. For instance if you had: A^v C_{\mu v} Wouldn't that just be: A^1C_{\mu 1} + A^2C_{\mu 2} + A^3C_{\mu 3} ? (this is one of the problems with a solution from Schutz's book and the solution starts...

Thanks for the help guys! I completely understand it in matrix form - for some reason I struggle with Einstein notation.

From Schutz's A First Course in General Relativity "A particle of rest mass m moves with velocity v in the x direction of frame O. What are the components of the four-velocity and four-momentum?" By definition \vec{U} = \vec{e}_{\bar{0} However, I don't see how he gets U^{\alpha} =...

This might be able to help: http://en.wikipedia.org/wiki/Glossary_of_tensor_theory#Classical_notation"

Okay, thanks, I'm pretty sure I understand this now. However, I'm probably going to have more questions as I continue through Schutz's treatment of the spacetime interval. Should I post them here, or make a new thread?

Is the following the correct expansion of: \Delta \overline{s}^2 = \sum_{\alpha = 0}^{3} \sum_{\beta = 0}^{3} M_{\alpha \beta} (\Delta x^{\alpha})(\Delta x^{\beta}) = \sum_{\alpha = 0}^{3} (M_{\alpha 0} \Delta x^{\alpha} \Delta x ^{0} + M_{\alpha 1} \Delta...

I have been working through Schutz's A First Course in General Relativity and was a little confused by how he presents the space time interval: \Delta \overline{s}^2 = \sum_{\alpha = 0}^{3} \sum_{\beta = 0}^{3} M_{\alpha \beta} (\Delta x^{\alpha})(\Delta x^{\beta}) for some...