We can explain the Schwarzschild metric with frame dragging

The gravitational wave from a binary black hole does have an undulating metric of time g₀₀ in the standard coordinates in which we calculate. But there is also undulating frame dragging because the wave is born from rapidly moving large masses.

The metric looks like the following:

             g₀₀        g₀₁        g₀₂        g₀₃

             g₁₀         .

             g₂₀                      .

             g₃₀                                     .

There is undulation on the row 0 and the column 0 in the matrix, as well as in g₁₁ and other purely spatial elements of the metric.

In literature it is shown that we can change the coordinates in such a way that g₀₀ becomes -1 and other elements on the row 0 and the column 0 become 0:

             -1            0            0           0

              0             .

              0                          .

              0                                         .

Thus, with a change of coordinates we show that there really was no undulation in the metric of time. But why we did have undulation when we were using our original coordinates?

The reason seems to be frame dragging. Frame dragging makes the time and space axes in our original coordinates non-orthogonal. That shows up, e.g., as a nonzero value of g₀₁. When we change the time coordinate to the proper time, so that g₀₀ becomes -1, we skew the coordinate axis of time relative to the spatial coordinate axes. It happens that this skewing makes the axes orthogonal again.

Frame dragging

It is as if the undulation in the metric of time in the original coordinates were due to frame dragging. In frame dragging, the "natural" inertial frame starts to move along with a large moving mass. If we want a clock to stay static relative to faraway observers, we have to exert a force on the clock. Otherwise, the clock would move along with the inertial frame.

Since we exert a force on the clock, it starts to move relative to the inertial frame and ticks slower.

If the black holes in the binary system move at a relativistuc speed, then gravitational waves become prominent. Also frame dragging is significant at such speeds.

Question. How does a faraway observer see the frame dragging that is included in a gravitational wave? The local observer inside the wave sees no undulation in the metric on the row 0 and the column 0. But can an outside observer see the undulation?

If a gravitational wave meets a very heavy object, then, obviously, frame dragging cannot drag that heavy object along. Then the distorted metric of time probably is revealed.

Time dilation and the stretching of the radial metric in the Schwarzschild solution can be explained with frame dragging

Let us drop a test mass from very far and let it fall toward a spherical mass.

Our freely falling test mass defines the "natural" inertial frame. An observer A who is static close to the mass is moving relative to the natural inertial frame. The clock of the observer ticks slower and the ruler that he uses to measure radial distances has become shorter.

We can explain the change in the clock rate and the length contraction of the ruler by the relative velocity v of the observer A relative to the natural inertial frame.

We can claim that the spherical mass "drags", or attracts, the natural inertial frame. Since the observer A is moving relative to that natural inertial frame, his clock appears to tick slower and the ruler is contracted.

This explains why the distortion of the metric in the Schwarzschild solution looks like the corresponding distortion for a rapidly moving object in special relativity. Time is slowed down by a factor γ and distances have grown by the factor 1 / γ in the direction of the movement.

There is a flaw, though, in our explanation. A clock attached to our freely falling test mass does not tick any faster than the clock of the observer A. The reason might be that the process of falling down is one-way. When a gravitational wave tugs on a test mass, the process is periodic. A clock floating freely in space will tick faster than a clock which we force to stay static in our initial coordinates.

A wave is "falling" in a highly distorted force field

If we wave an electric charge in our hand up and down, its Coulomb field produces electromagnetic waves far away. Far away, the electric field alternately points up and down.

We may claim that an opposite test charge far away is "falling" in a highly distorted Coulomb electric field of the charge in our hand. The test charge will mostly move up and down, but it will eventually reach the charge in our hand.

Our model may explain why some phenomena occur both in a free fall in a force field, and also if we let a test charge float freely inside a wave of the force field.

Conclusions

We found an explanation for the claim of literature that gravitational waves are purely spatial and that the metric of time does not undulate. Next we must find out how outside observers see the waves. In a black hole merger, the final phase lasts less than a second and produces a burst of gravitational waves. How do observers who are outside the burst see the passing of the burst? Can they measure it with precision clocks?

We were able to link the Schwarzschild metric to special relativity. That may help us in analyzing gravitational waves.