Bend it like dark matter!

By its very nature, dark matter can much motivate and engage students. Dark matter is a topic of modern physics, it features prominently in our understanding of the universe, and it presents scientists with unsolved problems. Research has shown that all these factors can have a positive impact on students' motivation and their attitudes towards science. For example, the large-scale survey ROSE (Relevance of Science Education) found topics of physics in space and astronomy to be popular among 15 year-old students [2]. The survey also found that students were interested in seemingly mysterious scientific phenomena or phenomena that scientists cannot yet explain. Other studies have corroborated these findings, which seem to be robust irrespective of age, gender or nationality [3, 4]. Inviting students to explore complex phenomena and unsolved problems in modern physics can provide productive learning opportunities [5–7].

Despite its mysterious nature, dark matter is a very accessible topic at the high school level. Teachers can link dark matter to many concepts that are already part of standard physics curricula such as gravity, circular and orbital motion, and optics. There are obvious links to topics of astronomy, particle physics, and general relativity that have started to become an increasing focus of curricula reforms in recent years [8, 9]. Hence, dark matter provides teachers with flexibility in choosing how to integrate the topic into existing practices.

Dark matter is an excellent example to illustrate 'science in the making' because there are many competing theories about the origins and the composition of dark matter. Teaching about dark matter can foster awareness of the tentative nature of scientific knowledge and illustrate aspects of scientific practices. There is a growing awareness in the physics education community that such familiarity with scientific methods enables students to contextualise their physics learning in a broader societal context [10, 11]. Thus, dark matter provides opportunities to embrace broader methodological, epistemic, and social perspectives of physics.

In summary, dark matter can serve as a hook to get students interested and engaged, it links to standard topics in physics curricula, and it allows teachers to incorporate perspectives of the nature of science.

In this paper, we focus on gravitational lensing as an instructive empirical clue of dark matter. As opposed to the classical empirical evidence of dark matter, gravitational lensing cannot be explained by classical physics alone. To make sense of this phenomenon, we need Einstein's general theory of relativity according to which massive objects such as stars and galaxies curve spacetime. Light propagates through spacetime along straight lines. Consequently, if the geometry of spacetime is curved, this affects the trajectories of light as well. Light continues to follow the straightest possible paths through curved spacetime which we interpret as the gravitational deflection, or gravitational lensing, of light (figure 1).

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The terminology of gravitational lenses stems from an analogy with ray optics. Just like optical lenses deflect light from its straight trajectory because of their refractive power, so does a 'gravitational lens' exert such a massive gravitational effect that the propagation of light is visibly influenced. Telescopes can capture such bending or distortions of light when massive objects act as gravitational lenses (figure 2).

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How do we know a gravitational lensing effect was caused by dark or ordinary matter? We do not! Indeed, for spacetime to curve, it does not matter whether the massive object is ordinary matter, dark matter, or even a black hole. Only the amount of mass and its distribution determines spacetime curvature and thus, the gravitational lensing effect. And this independence of the composition of matter has made gravitational lenses crucial in the search for dark matter because they provide an independent measurement of the mass of cosmic objects. For example, in addition to the classical evidence of dark matter, gravitational lenses offer a way of estimating the mass-to-light ratios of cosmic objects that uses assumptions of general relativity. The measurement of these ratios based on data from gravitational lenses has confirmed the classical missing-mass problem. Analyses of patterns of multiple lens images have also provided further clues about the actual distribution of dark matter in galaxies and galaxy clusters.

The amount of previous research on high school student conceptions of dark matter is minimal. However, Coble et al [15] reported that even undergraduate students were mostly unaware of the existence and origins of dark matter when entering a general education astronomy course. Moreover, even after instruction, students had difficulties in distinguishing between dark matter and dark energy. Finally, the authors suggest that students' maths skills, and in particular, the ability to read graphs, interfered with understanding the rotation curves of spiral galaxies as evidence for dark matter. This is an important finding since insufficient familiarity with reading graphs is likely to impede high school students' understanding of the subject as well. Based on numerous informal discussions with students and teachers that we have had in our role as physics educators, we wish to add another observation: Students often struggle to distinguish dark matter from dark energy but also from other 'mysterious' concepts such as antimatter and black holes [16].

In the context of gravitational lensing, students need to build understanding both of curved spacetime and light propagation in spacetime. Similar to the concept of dark matter, empirical evidence about high school students' understanding of spacetime is limited. First evidence suggests that high school students were intrigued by the idea of spacetime but struggled to visualise it [8, 9]. The four-dimensional nature of spacetime seemed to pose conceptual difficulties when students tried to conceptualise curvature or the geometry in four dimensions. Students were particularly bewildered about the time dimension and how time could be 'curved' or 'warped' [17]. We suggest using students' curiosity about spacetime as a hook to address our inability to visualise more than three dimensions. Encouragingly, high school students seemed to understand the concept of straight paths in curved spacetime pretty well. According to general relativity, light travels along straight trajectories (so-called geodesic curves) through spacetime. High school students appear to be able to characterise geodesic curves as the curved-spacetime generalisation of a straight line in various different but equivalent ways [18]. This finding is encouraging since it suggests that students can deal with the conceptual foundation of gravitational lensing.

In a study with high school and teacher training students, Galili and Hazan [19] described several patterns of reasoning in optics that deviated from a scientifically correct interpretation. For example, the authors referred to the 'flashlight scheme' if learners thought individual light rays were emitted from single points on the light source. Moreover, students tended to consider and draw only the relevant light rays emitted towards the target object (e.g. a lens) while ignoring light emitted in other directions. The authors referred to the 'image holistic scheme' if students interpreted an image as a material replication of an object. For example, some students argued that images could only be observed if they enter the eye of the observer.

Moreover, the 'spontaneous vision scheme' summarised student conceptions of seeing as a process that happens naturally in the presence of the eye. Indeed, some students thought that light was only needed to illuminate an object, or the eye itself. The students were often unaware of the role of light in the seeing process and the role of the eye as a photon detector.

Consequently, some students thought that light rays could be seen from the side, a misconception that is often supported by illustrations of lensing effects. Finally, for triangular prisms, students thought that the prism created two images by splitting up the light rays that originate from the light source. This misconception is particularly relevant for us since our jelly lens analogy builds on a triangular prism construction.

In this analogy, the laser pointer represents light from a distant galaxy. The laser pointer can be directed at the jelly lens at different angles to represent multiple light rays coming from the light source. The folded sheet of white paper represents the observer, for example, a telescope on Earth. The triangular jelly shape represents a gravitational dark matter lens, and the jelly itself represents curved spacetime.

If no jelly is placed between the laser pointer and the screen, light appears to travel in a straight line. However, if the laser light passes through the jelly, it changes direction. Similarly, if light passes through spacetime that is curved by dark matter, it looks like the light changes direction. In particular, light rays appear to be bent around dark matter clusters.

As described in section 2.2, some students think that we can see light beams from the side. Many visualisations of gravitational lensing use a ray model of light suggesting that we can see the light ray from the side as they are bent around a massive object. This might reinforce this misconception. Therefore, we recommend addressing the visibility of laser light beams, which is easily demonstrated with a laser pointer traversing air. We can only make a laser beam visible if it passes through a translucent medium, such as jelly, that forces some laser light to scatter into our eyes. Similarly, an observer on Earth cannot see the trajectories of light rays and will not know that the light passed through curved spacetime.

An observer cannot see the original light source without the lens. Therefore, assuming that light travels in straight lines, an observer will infer a different location of the light source based on what they can see: they would reconstruct an image of the light source. Similarly, an observer on Earth will infer a different location of a distant galaxy in case of gravitational lensing effects.

It can be challenging to understand the role of the observer and their position in the creation of images both for optical lensing and gravitational lensing. Therefore, we recommend letting students simulate different configurations of the light source, the lens and the observer, for example, by using the Ray Optics Simulation [30]. In this simulation, students can both create lenses of different shapes with varying refractive indices and change the light source, including the density of light rays and the position of the observer (figure 4). Working with this simulation, educators can explicitly address student conceptions related to the image reconstruction process, in particular, when using triangular lenses.

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Converging lenses are symmetrical and have a focal point, which leads to one sharp image of a light source. The further away from the optical axis light traverses the lens, the higher the refraction angle. Gravitational lenses, on the other hand, are usually not symmetrical and do not have a focal point. Instead, the further away from the centre of the dark matter mass light passes, the smaller the spacetime curvature and the smaller the bending effect on light. The index of refraction of space at a certain point is proportional to the gravitational potential there. Therefore, gravitational lenses have a focal line rather than a focal point. Consequently, gravitational lensing effects lead to displacement and distortions of the observed images, but only very rarely to magnifications. Indeed, in the early years of research on gravitational effects on light, authors used the term 'gravitational mirages' instead of 'gravitational lensing effect', in particular, if several images of a single source were produced [31].

In the jelly lens analogy, a triangular lens is used because it does not have a focal point either. Instead, the size of the angle of refraction only depends on the shape of the triangle. No matter how far away from the optical axis light traverses the jelly lens, the refraction angle will always be the same. The jelly lens is not an exact analogy to gravitational lensing; however, it does convey the main differences to a converging lens. Also, it is relatively easy to cut out triangular jelly shapes.

Depending on the distances between the light source, the lens, and the observer and their alignment, different images will be reconstructed, both for gravitational lenses and for jelly lenses. Here, simulations easily allow comparing different configurations. Einstein rings are one example of the effect of alignment. Suppose the light source, a point-mass gravitational lens, and an observer are perfectly aligned. In that case, the image of the light source appears as a circle of light, a so-called Einstein ring. However, if the gravitational lens is extended in size or the alignment is not perfect, the observers might see multiple, potentially distorted images.