## Gravity is an allusion? 🔗

The General Theory of Relativity tells us gravity is not a force, gravitational fields don’t exist. Objects tend to move on straight paths through curved spacetime.

### “Fun” math 🔗

#### Einstein field equations 🔗

#### general relativity 🔗

The basic principles of general relativity are:

● The geodesic postulate: free falling particles move along geodesics of space-time with the proper time $\tau$ or arc length $s$ as parameter. For particles with zero rest mass (photons), the use of a free parameter is required because for them holds $ds=0$. From $\delta\int ds=0$ the equations of motion can be derived:

● The *principle of equivalence*: inertial mass $\equiv$ gravitational mass $\Rightarrow$ gravitation is equivalent with a curved space-time were particles move along geodesics.

● By a proper choice of the coordinate system, it is possible to make the metric locally flat in each point $x_i$:

#### Riemann tensor 🔗

The *Riemann tensor* is defined as: $R^\mu_{\nu\alpha\beta}T^\nu:=\nabla_\alpha\nabla_\beta T^\mu-\nabla_\beta\nabla_\alpha T^\mu$, where the covariant derivative is given by $\nabla_j a^i=\partial_ja^i+\Gamma_{jk}^ia^k$ and $\nabla_j a_i=\partial_ja_i-\Gamma_{ij}^ka_k$.

Here, $\Gamma_{jk}^i=\frac{g^{il}}{2}\left(\frac{\partial g_{lj}}{\partial x^k}+\frac{\partial g_{lk}}{\partial x^j}-\frac{\partial g_{_jk}}{\partial x^l}\right)$, for Euclidean spaces this reduces to:

For a second-order tensor holds:

$[\nabla_\alpha,\nabla_\beta]T_\nu^\mu=R_{\sigma\alpha\beta}^\mu T^\sigma_\nu+R^\sigma_{\nu\alpha\beta}T^\mu_\sigma$, $\nabla_k a^i_j=\partial_ka^i_j-\Gamma_{kj}^la_l^i+\Gamma_{kl}^ia_j^l$, $\nabla_k a_{ij}=\partial_ka_{ij}-\Gamma_{ki}^la_{lj}-\Gamma_{kj}^la_{jl}$ and $\nabla_k a^{ij}=\partial_ka^{ij}+\Gamma_{kl}^ia^{lj}+\Gamma_{kl}^ja^{il}$.

The following holds: $R_{\beta\mu\nu}^\alpha=\partial_\mu\Gamma_{\beta\nu}^\alpha-\partial_\nu\Gamma_{\beta\mu}^\alpha+ \Gamma_{\sigma\mu}^\alpha\Gamma_{\beta\nu}^\sigma-\Gamma_{\sigma\nu}^\alpha\Gamma_{\beta\mu}^\sigma$.

#### Ricci tensor 🔗

The *Ricci tensor* is a contraction of the Riemann tensor:

The *Bianchi identities* are: $\nabla_\lambda R_{\alpha\beta\mu\nu}+\nabla_\nu R_{\alpha\beta\lambda\mu}+ \nabla_\mu R_{\alpha\beta\nu\lambda}=0$.

#### Einstein tensor 🔗

The *Einstein tensor* is given by:

With the variational principle $\delta\int(\ll(g_{\mu\nu})-Rc^2/16\pi\kappa)\sqrt{|g|}d^4x=0$ for variations $g_{\mu\nu}\rightarrow g_{\mu\nu}+\delta g_{\mu\nu}$ the *Einstein field equations* can be derived:

For empty space this is equivalent to $R_{\alpha\beta}=0$. The equation $R_{\alpha\beta\mu\nu}=0$ has as only solution a flat space.

The Einstein equations are 10 independent equations, which are of second order in $g_{\mu\nu}$. From this, the Laplace equation from Newtonian gravitation can be derived by stating: $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, where $|h|\ll1$. In the stationary case, this results in $\nabla^2 h_{00}=8\pi\kappa\varrho/c^2$.

The most general form of the field equations is:

$R_{\alpha\beta} - \frac{g_{\alpha\beta}R}{2}+\Lambda g_{\alpha\beta}=\frac{8\pi\kappa}{c^2}T_{\alpha\beta}$ where $\Lambda$ is the _cosmological constant_. This constant plays a role in inflationary models of the universe.### Video 🔗

## Your Thoughts? 🔗

Do you think that a freely falling object will radiate electromagnetic radiation or not? Are we simply all accelerating according to our interaction with matter (Earth in our case)? Until we can devise an experiment that can test this, we will only continue to guess and make assumptions.