# Tag Info

5

In relativity (both special and general) one of the key quantities is the proper length given by: $$ds^2 = g_{\alpha\beta}dx^\alpha dx^\beta \tag{1}$$ where $g_{\alpha\beta}$ is the metric tensor. The physical significance of this is that if we have a small displacement in spacetime $(dx^0, dx^1, dx^2, dx^3)$ then $ds$ is the total distance moved. You ...

3

Your last expression is not valid, for two reasons: first, any given index can only occur twice per term in the Einstein convention, once as an upper index and once as a lower index. Remember that when an index is repeated, it means you sum over it with the metric: $$T^a T_a = \sum_{a,b} g_{ab} T^a T^b$$ You have terms like ...

3

Take a trace of Einstein equations (trace of $g_{\mu \nu}$ is $D$), you obtain $$R - \frac{D}{2} R + D \Lambda = 0$$ Or $$R=\frac{D \Lambda}{D/2-1}$$ Then substitute this expression for $R$ into full Einstein equations and you obtain trivially $$R_{\mu \nu } = \frac{\Lambda}{D/2 - 1} g_{\mu \nu}$$

2

Definition 1. A spacetime is said to be spatially homogeneous if there is a one-parameter family of spacelike hypersurfaces $\Sigma_t$ foliating the spacetime such that for each $t$ and for any points $p,q\in\Sigma_t$ there is an isometry of the spacetime metric $g$ which takes $p$ to $q$. Definition 2. A spacetime is said to be isotropic if at each point ...

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John has already given a nice intuitive answer, I'd like to write it in a bit more mathematical way. First of all metric means a bilinear defined over a product (vector) space: $g(v,w): V \times V \rightarrow \mathbb{R}$. Where the vectors $v,w$ lie in the vector space $V$. This bilinear should be smooth, symmetric $g(v,w) = g(w,v)$, and positive definite , ...

2

The values of individual entries in the metric tensor depend on the coordinate system you choose. In the case of the FLRW metric there is a natural choice of coordinates called the comoving coordinates. In particular the comoving time has a very simple interpretation because it is equal to the proper time of a stationary observer, which obviously means ...

1

A pseudo Riemannian manifold is a manifold equiped with a metric of signature $(p,q)$, $p$ indicating the number of positive eigenvalues and $q$ the negative eigenvalues. For a Riemannian manifold, $q = 0$. A spacetime of dimension $n$ is defined by a pseudo-Riemannian manifold of signature $(1, n-1)$, or alternatively $(n-1,1)$, also called a Lorentzian ...

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Let's suppress some dimensions to simplify: $$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2$$ This quantity $$\Delta s^2$$ is preserved by changes of reference frame, just as in Galilean physics the quantity $$\Delta r^2 = \Delta x^2 + \Delta y^2$$ is preserved by rotations. Notice it is also the equation of a hyperbola. Thus, the effect of a frame shift is ...

1

A reference frame is equivalent to a choice of coordinates. So, choosing an accelerated frame in Minkowski space is equivalent to choosing a specific coordinate system on Minkowski space. Most importantly, this means that there is not genuine curvature in an accelerated frame, i.e. it is fundamentally different than gravity. The equivalence principle ...

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The boundary of a subset of a topological space is abstractly defined as the set-theoretic difference between its closure and its interior. Since topological spaces in general have neither coordinates nor metrics, this notion is independent of the metric. Since the spacetime manifold is a manifold, it is a topological space (locally homeomorphic to ...

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