All Objects are Already in Motion
General relativity, Albert Einstein’s theory, fundamentally reinterprets how gravity operates compared to classical Newtonian physics. In Newton’s view, gravity was a force acting at a distance between two masses. Einstein, however, proposed that gravity is not a force in the traditional sense but a manifestation of the curvature of spacetime caused by mass and energy.
In the framework of general relativity, all objects are already in motion (following their momentum and energy), unless acted upon by other forces (like electromagnetism, friction, etc.). This includes everything from planets to photons of light.
I repeat, objects are always moving, and what changes is the path of that motion due to the curvature of spacetime around them. When not influenced by other forces, they move along geodesics naturally due to their momentum.
When we talk about motion due to gravity in general relativity, we’re referring to objects moving along geodesics — the straightest possible paths in curved spacetime. An object moving through spacetime will continue to move along a geodesic unless acted upon by an external force.
The curvature is not something that pushes or pulls in a physical sense; it is a modification of the spacetime fabric itself, changing the definition of what “straight ahead” means for any object within that spacetime. This is a profound shift from the concept of gravity as a force acting at a distance.
The curvature of spacetime created by mass dictates the geodesics. This curvature doesn’t “push” or “pull” objects in the traditional sense. Instead, it shapes the path that objects naturally follow. The motion of objects under gravity in general relativity is akin to a ball rolling along a curved surface; the ball rolls in a certain way not because something is pushing or pulling it at each moment, but because the shape of the surface dictates its path.
What we perceive as gravitational attraction is really just bodies moving inertially (without any net force acting on them) in a curved spacetime environment. This inertial motion follows the spacetime curvature, which is influenced by mass and energy.
When an apple “falls” from a tree, from the apple’s own perspective — assuming no other forces like air resistance — it doesn’t feel like it’s moving. In its frame of reference, it’s following a geodesic, which is essentially the path of least resistance through spacetime. The apple is in a state of free fall, and thus, in a sense, it can be considered stationary relative to its immediate spacetime curvature.
When the apple falls, it’s simply following the natural curvature of spacetime created by the Earth’s mass. It is moving along a geodesic path, and this movement is the most natural state of motion in the presence of gravity. No external force is acting on the apple; it’s moving because that’s how spacetime guides it.
If no force is felt by the object (the apple, in this case), it doesn’t experience what we typically think of as acceleration, even though its velocity relative to the Earth changes. This is because in general relativity, moving along a geodesic is equivalent to being at rest or moving at a constant velocity in Newtonian mechanics.
From the apple’s perspective, it might consider itself stationary within its local frame of reference as it’s simply following the spacetime curvature without any force acting on it. However, an external observer on Earth sees the apple accelerating towards the ground due to gravity.
If we were to prevent the apple from falling, we would need to exert a force (such as holding it in our hand or placing it on a table). In this scenario, the apple would feel the force (the contact force from the hand or table), which would deviate it from its natural geodesic path. Thus, it would not be in free fall and would feel the force acting against the direction of the spacetime curvature.
True acceleration in general relativity is defined as a deviation from a geodesic path within spacetime. If an object is following a geodesic, it is in “free fall” and experiences no acceleration that can be felt internally — no “g-forces” are acting upon it. This is analogous to astronauts in orbit who are in continuous free fall around the Earth; they feel weightless even though they are accelerating towards the Earth in terms of changing their velocity direction.
When objects follow geodesics under the influence of gravity alone (without any other forces acting on them), they move in a way that might be described as “accelerating” by an external observer. For example, as a planet orbits a star, it continually changes direction, which is a form of acceleration. To the planet (or anything on it), this change in direction feels like no acceleration at all; it feels like moving straight ahead without any resistance or force.
The curvature of spacetime caused by a massive object (like a star or planet) determines the geodesics that other nearby objects will follow. When we see an apple falling to the ground, from a relativistic viewpoint, the apple is following a geodesic through curved spacetime created by Earth’s mass. The path is inherently curved so that it brings the apple towards the Earth. What we traditionally think of as “acceleration due to gravity” is actually the object following its natural path through spacetime, altered by Earth’s mass.
Mathematically, the presence and nature of acceleration are captured in general relativity by the metric tensor, which describes how spacetime is curved and how geodesics are shaped. Changes in an object’s velocity (speed and/or direction) as it follows a geodesic can be calculated through this tensor, indicating where and how “acceleration” occurs due to spacetime’s curvature.
This reversal of perspective is fundamental to understanding Einstein’s description of gravity. It removes the need for a gravitational “force” in the traditional sense and instead describes gravity as a geometric property of spacetime. The natural state of objects in this framework is to move along geodesics, and any deviation from this path (e.g., stopping the apple’s fall) requires an external force.
In general relativity, the tendency for objects to move from regions of less curvature to regions of more intense curvature in a gravitational context (such as an apple falling towards the Earth) is deeply rooted in how the theory interprets the effects of mass on spacetime. The reasons aren’t arbitrary; they stem from the fundamental properties of the spacetime geometry as influenced by mass and energy.
In curved spacetime, geodesics, which represent the paths of free-falling objects, naturally tend towards regions of greater gravitational influence, which are associated with more intense spacetime curvature. This happens because mass curves spacetime such that the “straightest” paths (geodesics) that objects can follow through spacetime lead towards the mass creating the curvature.
The Einstein Field Equations, which relate the spacetime metric (defining the geometry of spacetime) to the stress-energy tensor (representing the distribution of mass and energy), show that the curvature of spacetime (represented by the metric) increases as the density of mass-energy increases. This mathematical relationship is why the paths of objects curve towards regions of higher mass concentration.
The movement from less to more intense curvature areas is a manifestation of how spacetime itself behaves around masses according to general relativity’s predictions, which have been consistently confirmed by experiments and observations in physics. This behavior is not merely a convention but a result of the intrinsic properties of spacetime as influenced by mass and energy according to the theory’s fundamental equations.
Furthermore, in regions of stronger gravitational fields (more intense curvature), time is dilated more significantly compared to regions of weaker fields. This effect also influences the motion of objects through spacetime, effectively making paths that lead into stronger fields the natural trajectory under general relativity.
According to general relativity, the presence of a massive object can distort spacetime, leading to what is known as gravitational time dilation. In simple terms, clocks tick slower in stronger gravitational fields. This effect becomes more pronounced as the gravitational field strengthens (i.e., as you move closer to a massive object like a planet or a star).
Mathematically, this can be described by the Schwarzschild solution to Einstein’s field equations for a non-rotating spherical body. The closer you are to the source of gravity, the more significantly time is dilated. This is represented by the metric tensor of the spacetime, particularly how the element related to time changes with the gravitational potential.
Gravitational time dilation affects how objects move through spacetime because their motion is not only through space but also through time. In a region of spacetime where time is passing slower (near a massive object), the path of an object through spacetime (its worldline) is altered compared to regions where time passes more quickly.
In general relativity, an object’s path through spacetime is represented by its worldline. The worldline of the spaceship as it passes near a large mass is curved due to the gravitational influence of that mass. The curvature of this worldline is a representation of how spacetime itself is being curved by the mass.
As the spaceship’s worldline curves around the mass, the components of the spacetime metric affecting time change, resulting in time passing differently aboard the spaceship compared to areas outside of the mass’s influence. This change is not due to any mechanical or operational aspect of the spaceship but purely due to its movement through the curved spacetime created by the nearby mass.
In physics, particularly in the principle of least action, the path taken by an object between two points is the one that extremizes the action, which often translates into taking the least time. When considering the combined curvature of space and time, objects in a gravitational field will naturally follow paths that incorporate both spatial and temporal components of the curvature. This effectively means that the natural trajectory for an object in a gravitational field will be towards regions where time dilates more because these are the paths that extremize the action in the curved spacetime.
The time dilation effect can be calculated by considering the metric tensor of the spacetime around the mass, often using solutions like the Schwarzschild metric for spherical, non-rotating bodies. The metric describes how spacetime is stretched and time is dilated within this gravitational field. The formula typically involves the gravitational constant, the mass of the object creating the field, and the distance from this mass to the spaceship.
From a broader perspective, when an object has multiple potential paths through spacetime, the path that leads into stronger gravitational fields — and hence into regions of greater time dilation — often provides a trajectory that aligns with the natural geodesic paths determined by general relativity. This is because these paths are “shorter” in a four-dimensional spacetime sense, even if they are longer or slower in a purely three-dimensional spatial sense.
The trajectory of an object moving in a gravitational field, hence, is not merely about moving through space but also involves moving through time in a manner affected by the gravitational field. This comprehensive understanding of movement in both time and space underpins predictions such as the precession of the perihelion of Mercury and the bending of light near the sun, both of which have been experimentally verified.
In essence, gravitational time dilation is a critical factor that shapes the natural trajectories of objects under general relativity, emphasizing the unified nature of space and time in the theory. This phenomenon illustrates how deeply interconnected the fabric of spacetime is with the physical processes we observe, influencing everything from the orbits of planets to the paths of photons passing near a star.
References
Curved Space. Mathematics Department, Brown University. Retrieved from http://www.math.brown.edu/tbanchof/STG/ma8/papers/dstanke/Project/curved_space.html
SPACETIME!. Mathematics Department, Brown University. Retrieved from http://www.math.brown.edu/tbanchof/STG/ma8/papers/dstanke/Project/curved_space.html
