# Why gravitational wave stretches on $x$-axis would necessarily compress on the $y$-axis?

I read gravitational wave is a traverse wave, usually produced by inspiraling neutron stars or black holes, and laser interferometer such as LIGO is commonly used to detect them, since the setup of the mirrors of the interferometer is at 90° angle, and signal is read from the constructive and destructive patterns. Does it mean that due to the limitation of our detector we assume gravitational wave as being traverse wave? I think it depends on how the gravitational wave is generated, so inspiral binary stellar masses would always produce traverse wave. But is there any other way to produce longitudinal wave pattern such as heavy mass rocking in-situ, etc.?

• are you asking why the fabric of spacetime is rigid? – Árpád Szendrei Apr 6 '19 at 4:14
• Just curious since each point in spacetime ticks at different rate which depends on presence of mass, does it qualify as a field? Else what does it mean to excite a gravitational field in this sense?(optional) – user6760 Apr 6 '19 at 4:56
• The metric of spacetime, which is what determines its geometry, is the field. It has 10 values at each point in spacetime. The metric is the gravitational field. – G. Smith Apr 6 '19 at 5:40

## 2 Answers

There is no way to produce a longitudinal gravitational wave. They are inherently transverse.

To discover how gravitational waves behave, physicists mathematically study small perturbations to flat spacetime or some other fixed background spacetime. They do this by linearizing Einstein's field equations for General Relativity. The perturbations are found to propagate at the speed of light and to have only two modes, both of which are transverse to the direction of propagation.

The fact that, when a gravitational wave passes by, one transverse direction of space is found to expand while the other direction shrinks — and then vice versa, in an oscillatory way — is related to the fact that the metric field of spacetime is a tensor field with two indices.

It will be best if you try to understand the concept of polarization in light waves.

Light is described by a polarization, which is by convention the direction of the electric field as light propagates. Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. Note that the electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together.

This characterizes the light,

Light in the form of a plane wave in space is said to be linearly polarized. Light is a transverse electromagnetic wave, but natural light is generally unpolarized, all planes of propagation being equally probable. If light is composed of two plane waves of equal amplitude by differing in phase by 90°, then the light is said to be circularly polarized. If two plane waves of differing amplitude are related in phase by 90°, or if the relative phase is other than 90° then the light is said to be elliptically polarized.

Polarization is a vector, the direction of the electric field. At the quantum mechanical level this is explained by the fact that the photon, light emerges from zillion of photons, is a mass zero spin one particle and can only contribute to transverse polarization.

Gravitational waves also obey wave equations, their polarization is the direction of the gravitational field, but it is not a vector. The general relativity equations make them a transverse wave but the polarization is a tensor. So the effect of a passing gravitational wave will be affecting gravitational interactions in two different direction, the effect bound by the amplitude of the wave in a mathematically more complex manner, (it changes with the direction that is why it is a tensor), than for the electromagnetic polarization.It will always be transverse to its direction of motion. Linearly polarised gravitational wave

In a quantized gravitation (effective theories at present) the graviton is the equivalent of the photon and has mass zero and spin 2. Spin 2 is the reason for the gravitational wave built up by zillions of gravitons to have transverse polarization described by a tensor.

such as heavy mass rocking in-situ,

No, there is no way for longitudinal polarization. For a change in the gravitational field to produce a gravitational wave the mass accelerated has to have a quadrupole distribution, or the axis around which a mass distribution rotates should not pass through the center of mass for a non zero quadrupole moment.

• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Jun 28 '19 at 16:45