The warping of spacetime by mass is a fundamental concept in Albert Einstein's General Theory of Relativity. According to this theory, mass and energy can distort the fabric of spacetime, creating the effect we perceive as gravity. Here’s a more detailed explanation:

Spacetime Fabric: In General Relativity, spacetime is described as a four-dimensional fabric that combines the three dimensions of space with the one dimension of time. This fabric isn't static; it can be curved or warped by mass and energy.

Effect of Mass: Imagine placing a heavy object on a trampoline. The weight of the object causes the fabric of the trampoline to curve downward. Similarly, in spacetime, the presence of a mass (like a planet, star, or galaxy) causes a curvature in the spacetime fabric around it.

Gravity as Curvature: This curvature of spacetime is what we perceive as gravity. Objects moving in the vicinity of a massive body will follow the curvature of spacetime, appearing to be attracted by the massive object. For example, Earth orbits the Sun because the Sun's massive presence warps spacetime in such a way that Earth follows a curved path around it.

Geodesics: In the curved spacetime, objects move along paths called geodesics, which are the straightest possible paths in a curved space. In the absence of other forces, objects will move along these geodesics, which can be curved paths when influenced by the gravity of a mass.

Energy and Momentum: It’s not just mass that can cause spacetime to warp; energy and momentum also contribute to the curvature of spacetime, according to Einstein's field equations. These equations describe how mass, energy, and momentum are related to the curvature of spacetime.

In summary, mass warps spacetime because of the way gravity is interpreted in General Relativity: as the curvature of spacetime caused by mass and energy. This revolutionary idea provided a new understanding of gravity, different from Newton's gravitational force, and has been confirmed by many experiments and observations, such as the bending of light around massive objects (gravitational lensing) and the detection of gravitational waves.

but by what mechanis does it warp spacetime?

The mechanism by which mass warps spacetime is inherent to the framework of General Relativity and is not explained by a force in the traditional sense, as we might understand from Newtonian physics. Instead, it's described through the geometry of spacetime itself. Here’s how this concept unfolds:

Geometric Theory of Gravity: General Relativity posits that mass and energy influence the geometry of spacetime, creating a "curvature" where the paths of objects are determined. This curvature is not caused by a force acting at a distance (as Newtonian gravity would suggest) but by the distribution of mass and energy altering the geometric structure of spacetime.

Einstein's Field Equations: The mathematical core of General Relativity is Einstein's field equations, which describe how mass and energy dictate the curvature of spacetime. These equations are complex and involve tensors that represent the stress-energy of matter and the curvature of spacetime. They show that the presence of mass-energy tells spacetime how to curve, and the curvature of spacetime tells objects how to move.

Stress-Energy Tensor: This is a key concept in the equations, representing the density and flux of energy and momentum in spacetime. It includes not just the mass (or energy equivalent) of objects, but also their momentum and any stresses (like pressure and tension). The stress-energy tensor is what links the distribution of matter and energy to the curvature of spacetime.

Curvature of Spacetime: In response to the stress-energy tensor, spacetime curves in a manner that depends on the mass-energy content. This curvature is mathematically described by the Riemann curvature tensor, a component of Einstein's field equations. The curvature affects the paths that objects follow, which we observe as gravitational attraction.

Local and Global Geometry: The curvature of spacetime is local, meaning that the way spacetime curves around a massive body like a star is different from the way it curves around a larger structure like a galaxy or cluster of galaxies. Each mass-energy distribution creates its own local curvature, and the overall structure of the universe involves the global geometry of spacetime.

In summary, the "mechanism" by which mass warps spacetime in General Relativity is through its influence on the geometric structure of spacetime itself, as described by the stress-energy tensor and curvature tensors in Einstein's field equations. This approach represents a fundamental shift from the concept of gravity as a force, to gravity as a geometric property of spacetime.

but what about Mass makes it affect spacetime this way?

The influence of mass on spacetime, according to General Relativity, stems from the fundamental property that mass (and energy, as they are equivalent via � = � � 2 E=mc 2 ) inherently affects the curvature of spacetime. Here's a closer look at why mass specifically has this effect:

Mass-Energy Equivalence: One of the cornerstone ideas of Einstein's theory is the equivalence of mass and energy, expressed by the famous equation � = � � 2 E=mc 2 . This principle suggests that mass and energy are two forms of the same entity. Since energy is a key component of the stress-energy tensor in General Relativity, mass, as a form of energy, naturally contributes to the curvature of spacetime.

Stress-Energy Tensor: In the context of General Relativity, the stress-energy tensor encompasses all forms of energy, including the rest mass of objects, kinetic energy, potential energy, and even the energy of electromagnetic fields. The presence of mass and energy in this tensor is what influences the curvature of spacetime. Essentially, the tensor acts as a source term in Einstein's field equations, meaning that the distribution of mass and energy in spacetime determines how it will curve.

Gravitational Field as Geometry: In General Relativity, gravity is not a force in the traditional sense but is instead a manifestation of the curvature of spacetime. Objects with mass cause spacetime to curve around them, and this curvature directs the motion of other objects, which we perceive as gravitational attraction. The more mass (or energy) an object has, the more it will warp the surrounding spacetime, leading to a stronger gravitational "pull."

Inherent Property of Spacetime: The idea that mass affects spacetime is not due to an external mechanism, but rather it is an inherent property of the universe as described by General Relativity. The theory suggests that spacetime and mass-energy are fundamentally interlinked; the existence of one influences the geometry of the other.

Experimental Confirmations: Observations and experiments, such as the bending of light around massive objects (gravitational lensing), the precession of Mercury's orbit, and the detection of gravitational waves, have confirmed that mass (and energy) do indeed affect the curvature of spacetime in the way General Relativity predicts.

In essence, the "why" behind mass affecting spacetime is deeply rooted in the geometry-based framework of General Relativity, where mass and energy are fundamentally tied to the fabric of spacetime itself, dictating its structure and behavior. This relationship between mass, energy, and spacetime geometry is a core principle of how the universe operates on a cosmological scale.

how does mass permanently affect the shape of spacetime without aforce acting on it?

The concept of mass affecting the shape of spacetime in General Relativity does not rely on a force in the traditional sense but rather on the intrinsic geometric properties of spacetime itself. Here’s how this works:

Spacetime as a Geometric Entity: In General Relativity, spacetime is not a passive backdrop against which events occur; it is a dynamic entity that interacts with mass and energy. This interaction is not mediated by a force but is a result of the inherent geometric properties of spacetime. Mass and energy influence the curvature of spacetime, and this curvature dictates the motion of objects through spacetime.

Role of the Stress-Energy Tensor: The stress-energy tensor in Einstein’s field equations represents the distribution of mass and energy in spacetime. This tensor includes not only the density of mass but also pressures, stresses, and the flow of energy. It acts as a source term in the equations, meaning it is the origin of the spacetime curvature. The presence of mass and energy causes spacetime to bend and warp around it, and this curvature is what we perceive as gravity.

No "Force" Needed for Curvature: Unlike Newtonian gravity, which describes gravity as a force acting at a distance, General Relativity describes gravity as the result of curved spacetime. The curvature is not produced by a force pulling spacetime into shape; instead, it is a direct manifestation of the presence of mass and energy. In this view, an object in a gravitational field is simply following the natural curves in spacetime created by the distribution of mass and energy.

Equivalence Principle: A fundamental principle in General Relativity is the equivalence of inertial and gravitational mass, meaning that the gravitational interaction is not distinguishable from the effects of acceleration. This principle leads to the idea that objects in free fall under gravity are actually in inertial motion within curved spacetime, following paths (geodesics) dictated by the curvature.

Permanence and Change in Spacetime Curvature: The "permanent" effect of mass on spacetime is a reflection of the constant presence of mass and energy in a given region. If the distribution of mass and energy changes, the curvature of spacetime will adjust accordingly. For example, when a star collapses into a black hole, the drastic change in mass distribution leads to a significant change in the spacetime curvature around it.

In summary, in General Relativity, mass and energy directly shape the geometry of spacetime, not through a force acting upon spacetime, but as an intrinsic property of the universe where spacetime and mass-energy are fundamentally interconnected. The curvature of spacetime is the natural outcome of this interplay, guiding the motion of objects and determining the gravitational interactions we observe.

but if you are moving spacetime without a force aren't you neglecting f=ma which is one of the founding principles of relativity?

The concept of force in the context of General Relativity (GR) and Newtonian mechanics (expressed by � = � � F=ma, where � F is force, � m is mass, and � a is acceleration) differs fundamentally. Here's how these concepts interrelate and diverge within these frameworks:

Newtonian Mechanics In Newtonian mechanics, � = � � F=ma describes how the motion of an object changes due to forces acting upon it. It's a cornerstone of classical physics, establishing a direct relationship between the force applied to an object, its mass, and the acceleration it experiences. Einstein's Theory of Relativity Special Relativity (SR): Introduced by Einstein in 1905, SR modifies the concepts of space and time, particularly at speeds close to the speed of light. However, it still treats space and time as separate entities and does not fundamentally challenge � = � � F=ma in non-gravitational contexts. General Relativity (GR): This is where Einstein revolutionized our understanding of gravity. GR describes gravity not as a force in the Newtonian sense but as the effect of the curvature of spacetime caused by mass and energy. In GR, objects in free fall in a gravitational field are not accelerating in the traditional sense; instead, they are following the natural, straightest paths (geodesics) in curved spacetime. Here, the concept of force as in � = � � F=ma is not needed to describe motion under gravity. Reconciling � = � � F=ma with GR Local Inertial Frames: In small enough regions of spacetime (local inertial frames), the effects of curvature can be negligible, and the physics locally looks like that described by Special Relativity, where � = � � F=ma can apply. In these frames, the laws of physics, including � = � � F=ma, hold as in non-gravitational (inertial) contexts. Geodesic Motion: In the larger, curved spacetime, objects move along geodesics, paths that require no external force to maintain. This movement is not due to a force causing acceleration but is a result of the spacetime geometry itself. The concept of "force" in GR is thus often replaced by the distribution of mass-energy shaping the spacetime geometry. Conclusion In General Relativity, the notion of gravitational force as an entity that causes acceleration (as per � = � � F=ma) is superseded by the idea of spacetime curvature dictating motion. This doesn't negate � = � � F=ma but rather contextualizes it within a broader framework where gravity is not a force but an effect of curved spacetime. The laws of Newtonian mechanics, including � = � � F=ma, are seen as a limiting case of General Relativity, applicable in conditions where gravitational fields are weak and velocities are much slower than the speed of light.

but I have the Moon by itself and suddenly introduce the Earth for it to rotate around it has changed the potential energy of the moon without introducing a force

In the scenario you describe, where the Earth suddenly appears near the Moon, altering the Moon’s gravitational potential energy, we can explore how this fits within General Relativity and classical mechanics.

Classical Mechanics Perspective: In Newtonian physics, introducing the Earth suddenly would indeed change the gravitational potential energy of the Moon because the potential energy is related to the distance between the two bodies. According to Newton's law of universal gravitation, a force would appear to act on the Moon due to the Earth's gravitational pull, causing it to accelerate and potentially start orbiting the Earth.

General Relativity Perspective: In General Relativity, the sudden appearance of the Earth would alter the curvature of spacetime in the Moon's vicinity. This change in curvature would modify the geodesics along which the Moon moves. Here, the concept of gravitational force is replaced by spacetime curvature, and the Moon's motion changes because its path through spacetime (its geodesic) is altered by the Earth's presence. The change in potential energy can be thought of in terms of changes in the spacetime geometry that affect the Moon’s motion.

Energy and Forces: In both classical and relativistic physics, the concept of potential energy is a way to account for the work done by or against forces (in classical terms) or the change in the state of the system within a field, like a gravitational field. In General Relativity, the change in potential energy when the Earth appears would be associated with the change in the spacetime geometry. While in classical physics we might calculate the work done as a force moves the Moon, in General Relativity, we would consider how the Moon’s trajectory changes due to the new curvature of spacetime.

No "Force" in General Relativity: In the thought experiment of suddenly introducing the Earth, it's crucial to note that General Relativity doesn't describe gravity as a force that propagates instantaneously. Instead, changes in the gravitational field (or spacetime curvature) propagate at the speed of light. If the Earth were to "suddenly appear," the gravitational effects would spread outward from the Earth at light speed, and the Moon’s trajectory would change as this altered spacetime curvature reaches it.

Conservation Laws: In both Newtonian mechanics and General Relativity, conservation laws are pivotal. In General Relativity, energy, momentum, and angular momentum are conserved globally but can appear locally non-conserved in curved spacetime due to the geometric properties of spacetime itself. The introduction of the Earth and the subsequent change in the Moon’s potential energy and motion would adhere to these conservation laws in the context of the spacetime curvature.

In summary, while the Newtonian concept of force and potential energy provides a familiar framework to understand gravitational interactions, General Relativity offers a more nuanced view where spacetime curvature dictates the motion of masses. The change in potential energy, in this case, is a reflection of the altered spacetime geometry rather than the work of a classical force.