# Tagged Questions

52 views

### Riemann Curvature Tensor Symmetries Proof

I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies ...
16 views

108 views

### Why is the anti-symmetric tensor more important than symmetric tensors?

In differential geometry, the differential forms are anti-symmetric tensors. So, why is the anti-symmetric tensor like $d x_1 \otimes dx_2 - d x_2 \otimes d x_1$, more important than the ...
126 views

208 views

323 views

### Weight of a tensor density

Is there any freedom in choosing the weight of a tensor density? I have seen in some papers that they introduce a tensor density made from metric with a special weight. There is a tensor density with ...
1k views

### Riemann tensor in 2d and 3d

Ok so I seem to be missing something here. I know that the number of independent coefficients of the Riemann tensor is $\frac{1}{12} n^2 (n^2-1)$, which means in 2d it's 1 (i.e. Riemann tensor given ...
517 views

### Coordinate Transformation of Scalar Fields in QFT

By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu$. Correct? ...
256 views

### How do you show from the index notation that the change of frame formula for a metric must involve the transpose?

Let $x^\mu$ and $x^{'\mu}$ be two coordinate systems related by $$dx^{'\mu}~=~S^\mu{}_\nu~ dx^\mu.$$ In index notation the metric in both systems are related by: ...
217 views

### Weyl & Riemann curvature tensors and gravitational “physical” quantities in Einstein vacuum equations

If we look at the Einstein vacuum equations, that is without matter (there is the possibility or curvature without matter), for instance we may consider gravitational waves. The question is: Is there ...
221 views

### Metric tensor in General Relativity or otherwise [closed]

What is the metric tensor? How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices? How it relates to distance function (metric) and angles? How ...
145 views

### Tensors in general relativity

This is a question on the nitty-gritty bits of general relativity. Would anybody mind teaching me how to work these indices? Definitions: Throughout the following, repeated indices are to be summed ...
339 views

### Tensor equations in General Relativity

In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent. Question: why is this true? I'm looking for a ...
122 views

### Curvature and spacetime

Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
463 views

645 views

### Diffeomorphisms, Isometries And General Relativity

Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while. Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ ...
295 views

### Tensor Introduction

I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, ...