Recent content by vibhuav

I keep coming across this descriptor, "two (or three) independent, non-interacting parts," in many books on QM (for example, Penrose's Shadows of the Mind). It is usually followed by a mathematical description (for example, state vector |A>|B>). I can wrap my mind around the quantum paradox of...

Others have given excellent, statistics based, quite objective advice to which I can't add any more. However, I would like to give a "softer" and rather subjective point of view articulated best by American paleontologist, Stephen Gould, when he faced and won against cancer, at least the first...

I understand that the travelling twin (T, say) is subjected to acceleration and deceleration while the stay-at-home twin (S) is in inertial frame all the time. It is this asymmetry which results in the travelling twin aging less than the other, when they two meet up. Since acceleration is the...

Thanks for all the replies; I am still working it all out. In an attempt to understand how the metric of a 2D surface (even the tangent plane) embedded in 3D space still extols the non-flatness as the original metric equation did, I tried to consider the 1D “space” (basically a line) embedded...

Thanks for the replies. I will have to think through them carefully to understand completely. At this stage I would like to clarify my OP, quoted below: ...my main question is, I am not able to mathematically relate the original coordinates in 3D space to the coordinates of the tangent...

In trying to explain the concept of curved space, many books use the example of the surface of a sphere, which can be considered as a curved 2D space embedded in a higher dimensional, 3D space. I could derive, starting from ##a^2=x^2+y^2+z^2##, that the metric, or the line element, on the...

I think I understand now. First of all, let me correct a mistake that crept in. I now understand that ##\Delta r_{shell}## given in the OP is, in fact, the observed/measured distance between the 2 shells. For two shells with coordinate radii 10 and 11 km, k = 1.477, ##\Delta r_{shell}## turns...

By distance equation/distance formula, I was referring (hopefully correctly) to the spacetime interval which, around a Schwarzschild object, would be ##c^2d\tau^2 = A(r) c^2 dt^2 -\frac{dr^2}{A(r)} - r^2 d\theta^2 - r^2 \sin^2\theta d\phi^2##, from which ##\Delta r_{shell}## was derived

I wasn't quoting anybody, but my own understanding. Thanks anyway, i'll chew on your reply.

I tried to follow-up based on the replies I got, but I am not completely convinced. With the example in the OP, with ##k=1.477## (neutron star of ##1 M_\bigodot##) and ##r_1 = 10km, r_2=11km##, if I plug these coordinate radii into radial distance formula that I obtain by solving ## \int...

Many textbooks use the space (spacetime, actually, but for now only space is good enough) around a spherically symmetrical Schwarzschild object to demonstrate curvature of space due to gravity. Let’s consider two shells around such a Schwarzschild object (say a neutron star of 1 solar mass)...

OK, thanks, I think I get it. I also realized that I used "derived" when I meant "defined". A loose analogy would be the scalar, temperature, being defined (or described) as, say, "cold", but it derives a concrete value when you specify the units (or coordinates). In Fahrenheit it would be 68...

Can you please point me to some literature that does this? I would like to see, at least in principle, how the components of the metric tensor for spherical coordinates (for example) can be derived straight away without going through the Cartesian coordinate system.

I have to clarify that Fleisch's book (a student's guide to vectors and tensors) is one of the best introductory books on tensors; best explanations on contravarient and covarient components. It helped me very well.

Thanks to Orodruin and PeterDonis. I think my mistake was to take a specific example and generalize it. I'll have to be more careful, but thanks!