Free fall in curved spacetime—how to visualise gravity in general relativity

The first direct observation of gravitational waves in 2015 has led to an increased public interest in topics of general relativity (GR) and astronomy. Physics teachers and educators respond to this interest by introducing modern ideas of gravity and spacetime to high school students. Doing so, they face the challenge of finding suitable models that visualise gravity as the geometry of curved spacetime. Most models of GR, such as the popular rubber sheet model, only address spatial curvature. Yet, according to Albert Einstein, gravitational phenomena stem from deformations both in space and time. This paper presents a new model that builds on a relativistic generalisation of Newton’s first law. We use Einstein’s free fall thought experiment and a classical height-time diagram to explain how warped time gives rise to gravity. Our warped-time model acts as a convenient supplement to the rubber sheet model. To support teachers in integrating the model into their classroom practice, we have implemented the model as an interactive simulation that is freely accessible. The model is the result of a three-year period of developing and trialling digital learning resources in Norwegian high schools. Based on these trials, we suggest specific instructional strategies on how to use the warped-time model successfully in science classrooms.


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May 2019 space to explain planetary movement in an intui tive way, the model ignores deformations in time. Our warpedtime model presents an alternative strategy to explain gravity. The model thus acts as a useful supplement to the rubber sheet to visual ize how warped time makes objects fall.
The presentation of this paper follows a three fold structure: First, Einstein's key ideas on grav ity and spacetime are summarised by presenting two models of GR: the traditional rubber sheet model and our warpedtime model. The presenta tion lists advantages and limitations of each model as well. Second, the development of the warped time model are contextualised as part of the greater designbased research project ReleQuant that develops digital learning resources in mod ern physics [8]. Finally, the last section reports on students' experiences with the warpedtime model and discusses instructional implications to improve teaching and learning of GR.

Gravity and spacetime
This section summarises key ideas of GR and relates these ideas to two instructional models. The warpedspace model has become synony mous with GR, whereas the warpedtime model is our novel approach to visualising curved spacetime.

Warped-space model
At the heart of GR lies Einstein's field equa tion that describes the interplay between space, time, and massive objects [9]. The popular phrase 'spacetime tells matter how to move, matter tells spacetime how to curve' aptly encapsulates this equation [10]. The widely used rubber sheet model visualises this dynamic interplay through an intuitive handson activity [11].
The analogy compares the fabric of the uni verse to a stretched rubber sheet. Gravity is illus trated by placing a bowling ball and marbles on the rubber sheet. The bowling ball produces a warp of the rubber, which results in an inward tug that influences the movement of the marbles. It is the warp of the rubber sheet that creates the gravi tational tug. The rubber sheet model, sometimes also denoted spacetime simulator or pillow model [11,12], offers an intuitive explanation of gravity.
The deformed sheet provides a mechanism of how gravity arises and the model has great explana tory power: it is suitable to show orbital motions, curved space, and photon trajectories [13]. Yet, no instructional model comes without limitations. Research suggests that the rubber sheet might be misleading despite its visual power and simplic ity: The rubber sheet obscures that spacetime is 4D; in particular, the model obscures that space time has a temporal dimension [13].

Warped-time model
The warpedtime model addresses limitations of the rubber sheet model by offering a strategy to visualise gravity as an effect of warped time. The warpedtime model builds on another impor tant equation of GR, the geodesic equation. The geodesic equation is an equation of motion that can be thought of as a generalisation of Newton's first law. In an attempt to introduce the geodesic equation to science classrooms, physics educa tors recently coined the term 'Einstein's first law' [14]: Objects that are not influenced by forces move along geodesic curves in spacetime.
A geodesic curve is the spacetime generaliza tion of a straight line. The usefulness of geodesic curves in GR is that they are the paths followed by particles in free fall [15]. There is one important thing to note when formulating Einstein's first law: In contrast to classical mechanics, Einstein did not consider gravity to be a force. Thus, objects in free fall are indeed free-no force in the classical sense acts on them. Einstein's hap piest thought, namely that a person in free fall will experience a state of weightlessness, is an everyday example of Einstein's first law: Objects in free fall follow geodesic curves in spacetime.
Building on Einstein's first law, a new teach ing strategy makes the warping of time visible. The interactive warpedtime model is part of a digital learning environment in GR that is freely accessible at www.viten.no/relativity. The warpedtime model invites students to explore the physics of free fall both from a classical and from a relativistic perspective. As starting point, the model takes a digital heighttime diagram and presents students with two different scenarios tower. In the second case, he steps off the tower in line with his famous thought experiment. To familiarise students with the digital heighttime diagram, they are asked to draw trajectories into the heighttime diagram. This task serves as a warmup: Remaining on top of the tower corre sponds to a straight line in the heighttime dia gram and stepping off the tower corresponds to a parabola.
The second part of the warpedtime model shifts the two scenarios to a relativistic setting. This time, students have to take warped space time into account. Before they can draw trajec tories students have to move a slider to warp the timeaxis (figures 3 and 4). In this warped dia gram, remaining on top of the tower corresponds to a curved line and stepping off the tower cor responds to a straight line.
The difference between the classical height time diagram and its warped counterpart is that freefall trajectories either look curved or straight. Students learn that a straight path through spa cetime does not necessarily look like a 'straight line' in a given representation. Students learn to shift their perspective to understand that objects in free fall follow the straightest possible path through spacetime. Is it a force that pulls objects towards the ground? According to Einstein, there is no force pulling objects to the ground-it is the geometry of curved spacetime.
In a last step, the warpedtime model invites students to move between the Newton and Einstein models of gravity (figure 5). By moving a slider up and down, students can compare how both physicists explain the physics of free fall in two different ways: Newton treats gravity as a force that accelerates objects in free fall towards the centre of the Earth. The corresponding tra jectory in the spacetime diagram is a parabola. Einstein treats gravity as a geometric phenom enon. Objects in free fall follow geodesic curves in spacetime. In a warped heighttime diagram trajectories are straight indicating that there is no force acting on the object.
To help teachers use the warpedtime model successfully, it is important to list its strengths and limitations. One important limitation of the warpedtime model relates to the depiction of curvature. First, the warping of the timeaxis is greatly exaggerated. Relativistic effects of warped time are very small on the surface of the Earth [16]. Second, the curvature of the timeaxis is chosen in such a way as to make a freefall tra jectory in the heighttime diagram straight. Thus, the timeaxis curves somewhat arbitrarily and the curvature does not accurately correspond to the way spacetime is warped around the Earth 3 .
Another limitation of the warpedtime model relates to the double nature of gravity. The model does not distinguish between the two aspects of gravity that affect an object-one aspect due to acceleration and one part representing tidal forces. The free fall thought experiment demon strates the principle of equivalence: gravity and acceleration are locally indistinguishable: To describe a single idealised object in free fall one does not have to evoke curved spacetime expla nations. In this case, one can describe gravity by shifting to an accelerated frame of reference. Yet, in reality, objects have an extension and will experience tidal forces. Tidal forces arise from nonuniformities in the gravitational field and cannot be removed in free fall. These forces relate to spacetime curvature. A more thorough discus sion of tidal forces can be found in [17].
Despite its limitations, the warpedtime model has several strengths that make it an ideal supplement to spatial visualisations of GR: (1) The warpedtime model makes use of one of Einstein's most famous thought experiments and thought experiments are powerful tools to communicate relativistic concepts to high school students [18].

Educational context
The warpedtime model is the result of a design based research approach to developing learning resources in modern physics [6]. Topics of mod ern physics place high demands on students' understanding of abstract and often counterintu itive concepts. In response to these challenges project ReleQuant was established to study novel and innovative ways of teaching modern phys ics in Norwegian high schools [8]. In close col laboration with teachers and teacher students, the ReleQuant team developed a digital learning   viten.no/relativity In addition to having been developed within ReleQuant, the warpedtime model pools experi ence from EinsteinFirst and the Gravity Discovery Centre. EinsteinFirst is an Australian educational project that aims to introduce young learners to topics of relativity and quantum physics by developing simple models and handson activities [19]. The EinsteinFirst team coined the notion of 'Einstein's first law' in reference to the geodesic equation [14]. The Gravity Discovery Centre is an outreach facility and science museum colocated at the Australian International Gravitational Research Centre in Gin Gin, Western Australia. The centre features the socalled 'Leaning Tower of Gin Gin' which allows visitors to recreate free fall experiments [20]. The warpedtime model pre sented in this paper takes a digital version of the Leaning Tower as a setting to explore Einstein's law and freefall motion in curved spacetime.

Student experiences
The final design of the warpedtime model is a result of three iterative rounds of developing and testing learning resources in 12 Norwegian  physics classrooms. In this section, key insights from the classroom trials are summarised to guide instruction based on the warpedtime model. Generally, the classroom trials showed that students felt motivated and engaged by curved spacetime even though many admitted that the con cept was challenging [6]. The first trial of the learn ing resources suggested that students struggled to conceptualise movement along the timedimension [21]. The warpedtime model makes movement along the time dimension more visible for students by asking them to draw the trajectory of an object that remains spatially at rest. Understanding that objects always move in spacetime is an important insight that helps students integrate ideas of time and gravity into a relativistic framework.
The second trial of the learning resources targeted a prototype of the warpedtime activity specifically. Analysis of small group discussions showed that even though many students seemed to be comfortable with the idea of movement in space and time, only few groups were able to connect geodesic curves to the physical state of being in free fall [22]. Thus, successful instruction should aim to link the geometric description of GR to the physics of free fall. Focus group interviews sup ported the findings from the classroom discussions during the second trial. Students perceived a gap between relativistic and classical descriptions of free fall. Moreover, the interviews revealed that students continued to find it difficult to visualise time even though the warpedtime model helped them to get a better picture of this abstract concept.
Not all students approved of the warped time model though. Some criticised the model for not being representative of relativistic phe nomena. This criticism reveals an understanding of the limitations of this model as well as of the scope of Einstein's theory. Successful instruction of warped time should therefore complement the warpedtime model with other examples from cosmology and astrophysics where relativistic phenomena have a more significant effect.

Discussion and conclusion
Every instructional model has limitations. In learning domains such as GR where concepts are very abstract or impossible to visualise, it is crucial to develop different models that can complement each other [23]. This paper presents a new instructional model to visualise gravity as a manifestation of warped time. The model acts as a supplement to spatial models of GR such as the rubber sheet model. In addition to addressing the time dimension, the model introduces students to Einstein's first law.
Based on our classroom trials, we suggest four specific instructional strategies to use the warpedtime model successfully: (1) It is important to emphasise that every object moves both in space and in time. The insight that objects always move in time (in other words, they age) helps students link geometric descriptions of gravity to their everyday experience of gravity. (2) Einstein's thought experiment of freely falling objects is a popular introduction to GR. We suggest capitalising on this thought experiment and using the warpedtime model as a second step to explain gravity as a mani festation of curved spacetime. (3) In everyday life, relativistic phenomena cannot be observed directly. To help students make sense of warped time, we suggest using the warpedtime model to discuss gravita tional time dilation as well. (4) The warpedtime model allows for a direct comparison between the Newton and Einstein models of gravity by showing how two different models can describe the same physical phenomenon. Teachers should use this opportunity to help students build aware ness of the nature of scientific models.
The detection of gravitational waves and applications of GR create a fantastic vision of physics for the future. It is up to teachers and physics educators to bring this vision into sci ence classrooms. By offering novel instructional models that visualise gravity and curved space time, we hope to support teachers in engaging and inspiring the next generation of scientists.