Speeding through space: different velocities of gravitational wave polarizations

Gravitational waves are ripples in the fabric of spacetime that are constantly being emitted by extreme events throughout our universe. We can use the structure of these waves to study the objects that generated them and learn more about the true nature of gravity. For example, the pattern of stretching and squeezing of spacetime, called the polarization of the gravitational wave, can be used to distinguish different theories of gravity. In some theories, gravitational waves can have up to six different distinct patterns or polarizations. These different polarizations may travel at different speeds, which leads to intriguing observational consequences. Our recent paper, published in Physical Review D,  presents a new, theory-independent method for computing these polarizations with different propagation speeds, and the implications for gravitational physics.

Searching for these different polarizations is imperative to gravitational physics, as a direct detection of new polarizations could potentially lead us to a new theory of gravity. Such alternative theories may be necessary because although general relativity has passed every test to date, it leaves several open questions. One such question is: how can we reconcile general relativity with quantum mechanics? General relativity breaks down on very small scales; attempts to quantize gravity can lead to theories with polarizations that travel at different speeds. So far, only the two polarizations of general relativity have been observed. The discovery of any of the other possible polarizations could revolutionize our understanding of gravity. Conversely, if we do not find any additional polarizations, we can constrain alternative theories that predict these polarizations.

Measuring these waves with a gravitational wave detector is a complex task that requires us to accurately model their structure. In our work, we present a theory-independent method to compute these polarizations when they travel at different speeds. This generalizes influential previous work, which assumed all polarizations propagate at the speed of light. Our results are consistent with that work in the appropriate limit, validating our method. We also extend the parameterized post-Einsteinian formalism (which characterizes small modifications to general relativity in a theory-independent way) to theories with polarizations that have different propagation speeds. Finally, we discuss some of the observational consequences of different propagation speeds.

To develop this method, we first calculate the response of a detector to these different polarizations in full detail from the geodesic deviation equation (which describes how the detector is affected by a passing gravitational wave). We then use this result to create a more efficient method to calculate these polarizations through operators that act directly on the metric perturbation (a mathematical description of the small distortions in spacetime). This is a shorter, model-independent way to compute the gravitational wave polarizations. To demonstrate the effectiveness of our method, we use it in two specific theories of modified gravity, Einstein-æther theory and Khronometric gravity. Our expressions are consistent with other results for Einstein-æther theory and update the waves in Khronometric gravity - previously computed under the incorrect assumption that all polarizations in that theory propagate at the same speed. As the most general method to date, this is a valuable tool for future research. By extending current techniques and making the calculation of waveforms more efficient, our method allows for the study of other modified theories of gravity.

Yet, perhaps the largest contribution of our work is the discussion of observational consequences of polarizations that propagate at different speeds. We point out that different propagation speeds will result in different arrival times in the detector. Even a small difference in speed can build up over large distances, causing the waves to arrive as far as 10 years apart from each other, longer than we have so far observed gravitational waves. Thus, traditional techniques of searching for simultaneous polarizations may miss additional polarizations entirely, leading us to erroneously conclude that they do not exist. The traditional method of studying “null streams”, channels that would not contain any signal in general relativity, to rule out additional polarizations will not work for theories where those polarizations propagate at different speeds. Furthermore, if the speeds are different enough, additional polarizations may appear to be isolated, arriving without the two polarizations of general relativity. Therefore, our paper physically motivates novel searches for isolated additional modes.

Given our work's conclusions and the potential impact on the detectability of additional polarizations, we are extending this in ongoing research by proposing a novel search technique. In this new work, we suggest using the current constraints on propagation speed to reduce the portion of the data that we search for additional polarizations. This current project will examine in detail what constraints would be placed on the propagation speed of these additional modes if they are not detected.

This piece was based on the paper: Kristen Schumacher, Nicolas Yunes, and Kent Yagi. Gravitational wave polarizations with different propagation speeds. Phys. Rev. D, 108:104038, Nov 2023. (Available on the arXiv and Physical Review D.)