Could you help me with the earliest concept of Gravitational radiation, they say Einstein predicted it, but what exactly did he say or formulate? I tried to find it but came up with nothing other than that "he predicted it".
... I want to see what Einstein predicted.
Physics is about the confrontation of reliable, precise, and communicable frameworks of predicting domains of phenomena (physical models) with reality.
From 1910-1916 Einstein did the most significant work in his life and rewrote the rules of Newton's Universal Gravitation. The new theory, General Relativity, states that for any smooth, one-to-one space-time coordinate system with dimension 4, the proper time of any interval of any time-like world line is given as $$\tau = \frac{1}{c} \int \sqrt{ \sum_{\mu \in \{ 0,1,2,3 \} } \sum_{\nu \in \{ 0,1,2,3 \} } g_{\mu \nu}(x(\lambda)) \frac{d x^{\mu}(\lambda)}{d \lambda} \, \times \, \frac{d x^{\nu}(\lambda)}{d \lambda} } d\lambda $$ where $$x(\lambda)$$ is the position in space-time and $$g(x)$$ is a geometrical object known as a symmetric tensor. You can think of it as 10 functions of position in space-time where $$g_{\mu\nu}(x) = g_{\nu\mu}(x)$$. Einstein went on to say how $$g$$ relates to the density of energy, momentum and pressure at every point in space time, $$T$$.
Let $$\varepsilon^{\alpha \beta \gamma \delta} \equiv \varepsilon(\alpha, \beta, \gamma, \delta) = \frac{(\alpha - \beta)(\alpha - \gamma)(\alpha - \delta)(\beta - \gamma)(\beta - \delta)(\gamma - \delta)}{12}$$.
Let $$ \left| g(x) \right| \equiv \frac{1}{24} \sum_{\alpha \in \{ 0,1,2,3 \} } \sum_{\beta \in \{ 0,1,2,3 \} } \sum_{\gamma \in \{ 0,1,2,3 \} } \sum_{\delta \in \{ 0,1,2,3 \} } \sum_{\kappa \in \{ 0,1,2,3 \} } \sum_{\lambda \in \{ 0,1,2,3 \} } \sum_{\mu \in \{ 0,1,2,3 \} } \sum_{\nu \in \{ 0,1,2,3 \} } \varepsilon^{\alpha \beta \gamma \delta} \, \times \, \varepsilon^{\kappa \lambda \mu \nu} \, \times \, g_{\alpha \kappa}(x) \, \times \, g_{\beta \lambda}(x) \, \times \, g_{\gamma \mu}(x) \, \times \, g_{\delta \nu}(x) $$
Then $$\left( g^{\tiny -1} \right)^{\alpha \kappa}(x) = \frac{ \frac{1}{6} \sum_{\beta \in \{ 0,1,2,3 \} } \sum_{\gamma \in \{ 0,1,2,3 \} } \sum_{\delta \in \{ 0,1,2,3 \} } \sum_{\lambda \in \{ 0,1,2,3 \} } \sum_{\mu \in \{ 0,1,2,3 \} } \sum_{\nu \in \{ 0,1,2,3 \} } \varepsilon^{\alpha \beta \gamma \delta} \, \times \, \varepsilon^{\kappa \lambda \mu \nu} \, \times \, g_{\beta \lambda}(x) \, \times \, g_{\gamma \mu}(x) \, \times \, g_{\delta \nu}(x)}{ \left| g(x) \right| }$$
Such that $$ \sum_{\alpha \in \{ 0,1,2,3 \} } \left( g^{\tiny -1} \right)^{\alpha \mu}(x) \, \times \, g_{\alpha \nu}(x) = \delta_{\nu}^{\mu}$$ (where $$\delta$$ is the Kronecker delta on the right side) or $$ \sum_{\alpha \in \{ 0,1,2,3 \} } \sum_{\beta \in \{ 0,1,2,3 \} } \left( g^{\tiny -1} \right)^{\alpha \beta}(x) \, \times \, g_{\alpha \beta}(x) = \sum_{\alpha \in \{ 0,1,2,3 \} } \delta_{\alpha}^{\alpha} = \sum_{\alpha \in \{ 0,1,2,3 \} } 1 = 4$$
Let $$\Gamma_{\gamma \delta}^{\beta}(x) \equiv \Gamma(\beta, \gamma, \delta, x) = \frac{1}{2} \sum_{\alpha \in \{ 0,1,2,3 \} } \left( g^{\tiny -1} \right)^{\beta \alpha}(x) \, \times \, \left(
\frac{\partial g_{\alpha \gamma} (x) }{ \partial x^{\delta} } + \frac{\partial g_{\alpha \delta} (x) }{ \partial x^{\gamma} } - \frac{\partial g_{\gamma \delta} (x) }{ \partial x^{\alpha} } \right)$$
Let $$R_{\gamma \delta} (x) = \sum_{\alpha \in \{ 0,1,2,3 \} } \left( \frac{ \partial \Gamma_{\delta \gamma}^{\alpha}(x) }{\partial x^{\alpha} } - \frac{ \partial \Gamma_{\alpha \gamma}^{\alpha}(x) }{\partial x^{\delta} } + \sum_{\beta \in \{ 0,1,2,3 \} } \left( \Gamma_{\alpha \beta}^{\alpha}(x) \, \times \, \Gamma_{\delta \gamma}^{\beta}(x) \; - \; \Gamma_{\delta \beta}^{\alpha}(x) \, \times \, \Gamma_{\alpha \gamma}^{\beta}(x) \right) \right) $$
Let $$R (x) = \sum_{\beta \in \{ 0,1,2,3 \} } \left( g^{\tiny -1} \right)^{\alpha \beta}(x) \, \times \, R_{\alpha \beta}(x) $$
Then the "Einstein Field Equations" at the heart of General Relativity are:
$$R_{\alpha \beta} (x) \; - \; \frac{1}{2} R(x) \times g_{\alpha \beta}(x) \; + \; \Lambda \times g_{\alpha \beta}(x) = \frac{8 \pi G}{c^4} T_{\alpha \beta} (x)$$
Which is 10 coupled differential equations in $$g$$ and $$T$$.
Now I've never taken a course on GR, but I have read more than half of two textbooks on the subject and have assembled the above for your benefit from a variety of sources with some commonly suppressed summation signs and dependency on position spelled out. The whole point of taking a year (or more!) to teach GR is to work out the implications of this simple expression which is wildly non-linear in g.
Here's the simplest application of this:
If $$g_{00} = c^2 , \; g_{11} =g_{22} =g_{33} = -1 \, \; g_{01} = g_{02} = g_{03} = g_{12} = g_{13} = g_{23} = 0, \; \Lambda = 0$$
Then $$g^{\tiny -1} = g, \, \Gamma_{\gamma \delta}^{\beta} = 0, \, R_{\gamma \delta} = 0 \, R = 0 , \, \textrm{and} \, T = 0$$ so this is the metric of flat, empty space, expressed in Cartesian coordinates and so $$\tau = \frac{1}{c} \int \sqrt{ c^2 \left( \frac{d x^{0}(\lambda)}{d \lambda} \right)^2 - \left( \frac{d x^{1}(\lambda)}{d \lambda} \right)^2 - \left( \frac{d x^{2}(\lambda)}{d \lambda} \right)^2 - \left( \frac{d x^{3}(\lambda)}{d \lambda} \right)^2 } d\lambda = \frac{1}{c} \int \sqrt{ c^2 - \left( \frac{d X(t)}{d t} \right)^2 - \left( \frac{d Y(t)}{d t} \right)^2 - \left( \frac{d Z(t)}{d t} \right)^2 } dt = \int \sqrt{ 1 - \left( \frac{\vec{v}(t)}{c} \right)^2 } dt$$ exactly as in Special Relativity.