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 ss as parameter. For particles with zero rest mass (photons), the use of a free parameter is required because for them holds ds=0ds=0. From δds=0\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 xix_i:

Riemann tensor 🔗

The Riemann tensor is defined as: RναβμTν:=αβTμβαTμR^\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 jai=jai+Γjkiak\nabla_j a^i=\partial_ja^i+\Gamma_{jk}^ia^k and jai=jaiΓijkak\nabla_j a_i=\partial_ja_i-\Gamma_{ij}^ka_k.

Here, Γjki=gil2(gljxk+glkxjgjkxl)\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:

[α,β]Tνμ=RσαβμTνσ+RναβσTσμ[\nabla_\alpha,\nabla_\beta]T_\nu^\mu=R_{\sigma\alpha\beta}^\mu T^\sigma_\nu+R^\sigma_{\nu\alpha\beta}T^\mu_\sigma, kaji=kajiΓkjlali+Γkliajl\nabla_k a^i_j=\partial_ka^i_j-\Gamma_{kj}^la_l^i+\Gamma_{kl}^ia_j^l

, kaij=kaijΓkilaljΓkjlajl\nabla_k a_{ij}=\partial_ka_{ij}-\Gamma_{ki}^la_{lj}-\Gamma_{kj}^la_{jl} and kaij=kaij+Γklialj+Γkljail\nabla_k a^{ij}=\partial_ka^{ij}+\Gamma_{kl}^ia^{lj}+\Gamma_{kl}^ja^{il}.

The following holds: Rβμνα=μΓβνανΓβμα+ΓσμαΓβνσΓσναΓβμσR_{\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:

Rαβ:=RαμβμR_{\alpha\beta}:=R^\mu_{\alpha\mu\beta}, which is symmetric: Rαβ=RβαR_{\alpha\beta}=R_{\beta\alpha}.

The Bianchi identities are: λRαβμν+νRαβλμ+μRαβνλ=0\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:

Gαβ:=RαβgαβR2G^{\alpha\beta}:=R^{\alpha\beta}-\frac{g^{\alpha\beta}R}{2}, where R:=RααR:=R_\alpha^\alpha is the _Ricci scalar_, for which holds: βGαβ=0\nabla_\beta G_{\alpha\beta}=0.

With the variational principle δ((gμν)Rc2/16πκ)gd4x=0\delta\int(\ll(g_{\mu\nu})-Rc^2/16\pi\kappa)\sqrt{|g|}d^4x=0 for variations gμνgμν+δgμνg_{\mu\nu}\rightarrow g_{\mu\nu}+\delta g_{\mu\nu} the Einstein field equations can be derived:

Gαβ=8πκc2Tαβ\displaystyle G_{\alpha\beta}=\frac{8\pi\kappa}{c^2}T_{\alpha\beta} , which can also be written as Rαβ=8πκc2(TαβgαβTμμ2)R_{\alpha\beta}=\frac{8\pi\kappa}{c^2}(T_{\alpha\beta}- \frac{ g_{\alpha\beta}T^{\mu}_{\mu}}{2})

For empty space this is equivalent to Rαβ=0R_{\alpha\beta}=0. The equation Rαβμν=0R_{\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 gμνg_{\mu\nu}. From this, the Laplace equation from Newtonian gravitation can be derived by stating: gμν=ημν+hμνg_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}, where h1|h|\ll1. In the stationary case, this results in 2h00=8πκϱ/c2\nabla^2 h_{00}=8\pi\kappa\varrho/c^2.

The most general form of the field equations is:

RαβgαβR2+Λgαβ=8πκc2TαβR_{\alpha\beta} - \frac{g_{\alpha\beta}R}{2}+\Lambda g_{\alpha\beta}=\frac{8\pi\kappa}{c^2}T_{\alpha\beta} where Λ\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.