7.2 Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity
3 min read•june 24, 2024
describes how objects attract each other due to their mass. It's a simple yet powerful formula that explains everything from falling apples to planetary orbits.
While Newton's law works for most everyday situations, provides a more accurate description of gravity in extreme conditions. It views gravity as a warping of rather than a force.
Newton's Law of Universal Gravitation
Newton's gravity vs Einstein's relativity
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- Newton's law of universal gravitation describes gravity as a force proportional to the masses of objects and inversely proportional to the square of their separation distance ()
- Assumes gravity acts instantaneously between objects ()
- Treats gravity as a force distinct from other forces (electromagnetism, strong nuclear force, weak nuclear force)
- Works well for most practical applications (planetary orbits, projectile motion) but breaks down in extreme conditions (black holes, high-speed motion)
- Einstein's theory of general relativity describes gravity as a consequence of the caused by the presence of mass and energy
- Predicts that the effects of gravity propagate at the speed of light () rather than instantaneously
- Treats gravity as a geometric property of spacetime rather than a distinct force
- Provides a more accurate description of gravity, especially in strong gravitational fields (near black holes) and at high velocities (approaching the speed of light)
- Reduces to Newton's law of universal gravitation in weak gravitational fields and at low velocities (everyday experience on Earth)
Calculations with Newton's gravity
- Newton's law of universal gravitation expressed mathematically as
- represents the gravitational force between two objects
- is the with a value of
- and are the masses of the two objects
- is the distance between the centers of the two objects
- can be calculated using
- represents the gravitational acceleration of an object
- is the mass of the larger object exerting the gravitational force (Earth, Sun)
- is the distance between the centers of the objects
Gravity's effects on celestial objects
- Orbits of planets and satellites result from the balance between gravitational force and the object's velocity
- Gravity causes celestial bodies to follow elliptical orbits around more massive objects (planets around the Sun, moons around planets)
- The shape and size of the orbit depend on the masses of the objects and their initial velocities (, )
- General relativity predicts a small precession of orbits, particularly noticeable in the of Mercury
- Tides on Earth are caused by the gravitational forces from the Moon and Sun
- The difference in gravitational force across Earth's diameter leads to tidal bulges (high tides) on opposite sides of the planet
- The relative positions of the Earth, Moon, and Sun determine the strength of the tides ( vs )
- is a consequence of general relativity where massive objects curve spacetime, causing light to follow curved paths near them
- This effect can create multiple images of a single object (quasars), magnify distant objects (galaxies), or distort the apparent shape of objects (arcs, rings)
- is another consequence of general relativity where clocks in stronger gravitational fields run slower compared to clocks in weaker gravitational fields
- This effect is measurable and must be accounted for in GPS satellites to ensure accurate positioning on Earth's surface
General Relativity Concepts
- The states that the effects of gravity are indistinguishable from the effects of acceleration in a small region of spacetime
- In general relativity, free-falling objects follow , which are the straightest possible paths in curved spacetime
- The presence of mass and energy causes the curvature of spacetime, which in turn determines how objects move
- The is the distance from the center of a very compact object at which the escape velocity equals the speed of light, defining the event horizon of a black hole
Key Terms to Review (24)
$a = \frac{F}{m} = G \frac{M}{r^2}$: $a = \frac{F}{m} = G \frac{M}{r^2}$ is a fundamental equation that relates the acceleration of an object to the forces acting upon it, specifically the force of gravity. This equation is central to both Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity, as it describes the relationship between an object's mass, the gravitational force acting on it, and its resulting acceleration.
$F = G \frac{m_1 m_2}{r^2}$: $F = G \frac{m_1 m_2}{r^2}$ is the mathematical expression that describes Newton's Law of Universal Gravitation, which states that any two objects with mass will exert a gravitational force on each other. This force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This formula is also relevant in the context of Einstein's Theory of General Relativity, which provides a more comprehensive understanding of gravity as a consequence of the curvature of spacetime.
$G$: $G$ is a fundamental constant in physics that represents the strength of the gravitational force between two objects. It is a crucial parameter in both Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity, as it quantifies the attractive force between masses and governs the behavior of gravitational fields.
Action at a Distance: Action at a distance refers to the ability of one object to exert a force on another object without any physical contact or intermediary medium between them. This concept is central to the understanding of gravitational and electromagnetic interactions in physics.
Albert Einstein: Albert Einstein was a renowned German-born theoretical physicist who developed the theories of special and general relativity, which revolutionized our understanding of space, time, and the universe. His groundbreaking work also had significant implications for fields such as Newton's Law of Universal Gravitation and the quantum nature of light.
Curvature of Spacetime: The curvature of spacetime is a fundamental concept in Einstein's Theory of General Relativity, which describes gravity as a consequence of the distortion or curvature of the four-dimensional spacetime continuum. This curvature is caused by the presence of mass and energy, and it affects the motion of objects within that spacetime.
Eccentricity: Eccentricity is a measure of the degree to which an elliptical orbit deviates from a perfect circle. It is a dimensionless quantity that describes the shape of an ellipse, with a value ranging from 0 for a perfect circle to a value greater than 0 and less than 1 for an ellipse. This concept is crucial in understanding the motion of celestial bodies and the laws governing their orbits.
Einstein's Theory of General Relativity: Einstein's Theory of General Relativity is a fundamental theory in physics that describes gravity not as a force, but as a consequence of the curvature of spacetime caused by the presence of mass or energy. It is a comprehensive theory that provides a unified description of gravity as a geometric property of space and time.
Equivalence Principle: The equivalence principle is a fundamental concept in Einstein's theory of general relativity, which states that the effects of gravity are indistinguishable from the effects of acceleration. This principle forms the foundation for understanding the relationship between gravity and the curvature of spacetime.
Geodesics: Geodesics are the shortest paths between two points on a curved surface, such as the surface of a sphere or a curved spacetime. They represent the natural trajectories of objects moving under the influence of gravity or other curved geometries.
Gravitational Acceleration: Gravitational acceleration is the acceleration experienced by an object due to the Earth's gravitational pull. It is a fundamental concept in both Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity, describing the rate at which the velocity of an object changes when it is subjected to the force of gravity.
Gravitational Constant: The gravitational constant, denoted as 'G', is a fundamental physical constant that describes the strength of the gravitational force between two objects. It is a crucial parameter in the mathematical formulation of Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity, which govern the motion of celestial bodies and the curvature of spacetime.
Gravitational Lensing: Gravitational lensing is a phenomenon where the gravitational field of a massive object, such as a galaxy or a cluster of galaxies, bends and distorts the path of light traveling through it. This effect was predicted by Albert Einstein's theory of general relativity and has become an important tool in the study of astrophysics and cosmology.
Gravitational Time Dilation: Gravitational time dilation is a phenomenon predicted by Einstein's theory of general relativity, where the passage of time is affected by the presence of gravitational fields. This means that time flows at different rates depending on the strength of the gravitational field an object is experiencing.
Gravitational Waves: Gravitational waves are distortions in the fabric of spacetime, propagating at the speed of light, that are generated by the acceleration of massive objects. These waves are a prediction of Einstein's theory of general relativity and have been directly observed, providing experimental confirmation of this fundamental aspect of the theory.
Inverse Square Law: The inverse square law is a fundamental principle that describes the relationship between the strength or intensity of a physical quantity and the distance from the source of that quantity. It states that the strength or intensity of a physical quantity is inversely proportional to the square of the distance from the source.
Isaac Newton: Isaac Newton was an English mathematician, physicist, astronomer, and natural philosopher who is widely regarded as one of the most influential scientists of all time. His groundbreaking work in the fields of mechanics, optics, and astronomy laid the foundation for our modern understanding of the physical world.
Neap Tides: Neap tides are a type of tide that occurs when the gravitational pull of the sun and moon are perpendicular to each other, resulting in a smaller difference between high and low tides. This phenomenon is directly related to Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity, which describe the gravitational forces that influence the Earth's tides.
Newton's Law of Universal Gravitation: Newton's Law of Universal Gravitation is a fundamental principle in physics that describes the gravitational force between any two objects with mass. It states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Perihelion Precession: Perihelion precession is the gradual shift in the orientation of the elliptical orbit of a planet or other celestial body around the Sun. This phenomenon occurs due to the combined effects of Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity, which describe the gravitational interactions between celestial bodies.
Schwarzschild radius: The Schwarzschild radius is a critical distance around a massive object, such as a black hole, within which the object's gravitational pull is so strong that not even light can escape. It represents the point of no return, where the gravitational force becomes so intense that it creates an event horizon, a boundary beyond which information cannot be transmitted to the outside world.
Semi-Major Axis: The semi-major axis is a fundamental parameter that defines the size and shape of an elliptical orbit. It represents the length of the longest radius of the ellipse, or the average distance between an orbiting body and the object it is orbiting.
Spacetime: Spacetime is a unified mathematical model that combines the three dimensions of space and the one dimension of time into a single four-dimensional continuum. This concept is fundamental to the understanding of physics, particularly in the context of Einstein's theories of relativity.
Spring Tides: Spring tides are exceptionally high and low tides that occur when the gravitational pull of the sun and moon align, creating a stronger combined gravitational force that results in more extreme tidal ranges. This phenomenon is closely linked to Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity, which describe the fundamental forces governing the motion and interactions of celestial bodies.