Why is spacetime curved? - by Hossein Javadi, Mark John Fernee and Viktor T. Toth - Quora Question Review

This document contains a review of the answer by Hossein Javadi, Mark John Fernee and Viktor T. Toth on the question in Quora: "Why is spacetime curved?"
To order to read all the answers select: https://www.quora.com/Why-is-spacetime-curved?

• The text in italics is copied from the article.
  
• Immediate followed by some comments
  

Contents

• 1. Answer Review by Hossein Javadi
• 2. Answer Review by Mark John Fernee
• 3. Answer Review by Viktor T. Toth

Reflection

• Reflection 1 - Question Review
• Reflection 2 -

1. Answer Review by Hossein Javadi

In 1854, Riemann delivered his monumental lecture “Uber die Hypothesen, welche der Geometrie zu Grunde liegen”. In this lecture, he described how to generalize Gauss’ idea of surfaces and their curvatures to higher dimensions and thus singlehanded createdly the subject of Riemannian geometry. Riemann discusses various possibilities by means of which an n-dimensional manifold may be endowed with a metric, and pays particular attention to a metric defined by the positive square root of a positive definite quadratic differential form.

In 1907, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity. Riemann’s work was so ahead of his time that it did not receive its proper attention in the mathematical community until Einstein’s formulation of general relativity in 1915. Einstein imagined the spacetime as a geometric object whose curvature is determined by the distribution of energy and matter. Thus gravitational force is no longer a force in the Newtonian sense but a mere manifestation of the curvature of spacetime . The space-time is a manifold and a space-time event is represented by a point . The worldline of the particle is traced by a curve in . There are several basic tenets of general relativity. • Space-time is a semi-Riemannian manifold. • Free particles follow geodesics. • Curvature tells matter how to move, and matter tells space-time how to curve. Intrinsic properties - those that can be measured on the surface without regard to how it is embedded in an ambient space. In Euclidean geometry, the shortest distance between two points can be found using Pythagoras's theorem. What Riemann discovered was a more powerful, general form of Pythagoras's theorem that works on curved surfaces, even when the curvature is in more than two dimensions and varies from one place to another. In this looking-glass world of curved space, the familiar idea of distance is replaced by the broader concept of something called a metric, from the Greek for "measure," while curvature is similarly described by a more elaborate mathematical object. Gauss had found that the curvature in the neighborhood of a point of a specified two-dimensional geometry is given by a single number: the Gaussian curvature. Riemann showed that six numbers are needed to describe the curvature of a three-dimensional space at a given point, and that 20 numbers at each point are required for a four-dimensional geometry: the 20 independent components of the so-called Riemann curvature tensor.

How are most of these components measured? If not all measured: How are they calculated?

1.1. Principle of Equivalence

Experiments performed in a uniformly accelerating reference frame with acceleration a are indistinguishable from the same experiments performed in a non-accelerating reference frame which is situated in a gravitational field where the acceleration of gravity = g = -a = intensity of gravity field.

Before of Einstein, Hilbert had submitted a paper the foundations of physics which also contained the correct field equations for gravitation.

1.2. Einstein field equation

The Einstein field equations is the set of 10 equations that describe the fundamental interaction of gravitation as a result of space-time being curved by mass and energy. These equations are used to study phenomena such as gravitational waves.

General relativity is the geometric theory of gravitation and the current description of gravitation in modern physics. In general relativity, the universe has three dimensions of space and one of time and putting them together we get four dimensional spacetime, which gravity as an emergent effect from the spacetime curvature associated with distributions of energy. The central ideas of general relativity have been neatly summarized by John Archibald Wheeler: "matter tells space how to bend; space tells matter how to move".

Energy cause space-time warp and curve and gravity is the exchange of gravitons. As the photon is packet of electromagnetic energy, gravitons would be considered packets of the gravitational energy or space-time curvature.

2. Answer Review by Mark John Fernee

The reason that space-time must be curved in General Relativity follows from a couple of insights and centres on Einstein's self-professed greatest idea.

One of the key requirements for the formulation of General relativity was the principle of equivalence between inertial mass and gravitational mass.

It may not be so clear why this step is important until you understand that Special relativity is formulated with inertial mass, so to extend the Special theory to the General theory, it is necessary to suppose that there is no difference between a mass that is acted on by an external force and a mass that is acted on by gravity.

Einstein's great insight was that a body in free-fall in a gravitational field has no external forces acting on it.

The speed of such a body will increase.

This is similar to an inertial frame in Special relativity, that maintains a constant velocity in the absence of an external force.

What is the definition of inertial frame?

So an inertial trajectory in the absence of gravity is indistinguishable from a free-falling trajectory in a gravitational field.

This can only be explained by means of an experiment, described in detail.

Thus a free-falling trajectory can be thought of a generalisation of an inertial trajectory, but now the path traced out will be curved due to gravitational accelerations.

The important thing with the General theory is that it incorporates the Special theory. That means that mass-energy equivalence must be considered. This means that light, which has energy, should also respond to a gravitational field. Thus light will follow a free-falling trajectory, which is referred to as a null-geodesic.

The difficulty now is that if light can follow curved paths, these curved paths must be the path of least time between two points. The only way that the curved path can be the path of least time and the speed of light remain constant is that if space and time are compressed and stretched respectively.

That means that (the behaviour) light does not follow standard physical behaviour.

This is the curvature of General relativity. It's not so hard to show that time must run slower and space be compressed to maintain the constant speed of light in a curved trajectory.

This explains

So the essential parts in understanding that space-time is curved are that light takes free-falling trajectories and that the speed of light must be constant. Just the realisation that light takes curved paths near gravitating objects was the major new insight that made Einstein a household name.

3. Answer Review by Viktor T.Toth

You stumbled upon the reason why Einstein himself was never comfortable with the notion that “gravity is just curvature”: why in his public and private correspondence, never stopped referring to gravity as a force, and always referred to the geometric view as some kind of a “mental aid”, not necessarily a good description of reality.

But allow me to take a few steps back, all the way to Newton, who first described the gravitational relationship between two bodies using mathematics. The idea was simple: the attractive gravitational force is proportional to the masses of the two bodies and inversely proportional to the square of the distance between them.

That is a mathematical explanation. What you need is a physical explanation.
A much better question to answer is: What causes two object to attract each other.
Maybe an even better question is: What is involved to separate two objects.

But then, he ran into a philosophical issue. What mediates gravity?

In a letter to Richard Bentley, he wrote, “[t]hat gravity should be innate inherent & essential to matter so that one body may act upon another at a distance through a vacuum without the mediation of any thing else by & through which their action or force may be conveyed from one to another is to me so great an absurdity that I believe no man who has in philosophical matters any competent faculty of thinking can ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws, but whether this agent be material or immaterial is a question I have left to the consideration of my readers.”

Laws cannot be used as an explanation.

Of course he wrote this nearly two centuries before the concept of a field, such as the electromagnetic field, emerged in physics: a field that has a material existence in that it contains and carries energy and momentum, and importantly, mediates influences between bodies at a finite speed.

The most important concept is: That a field mediates influences between bodies at a certain speed.
This raises a more important issue: How does such an influence physical operates.
The physical issue is that a field is the 'same' as a quantum of force or an unit of force.

I am, of course, referring to Maxwell’s theory of the electromagnetic field, which finally came to its own right as a fundamental entity, freed from any material shackles in the form of imaginary stuff like the luminiferous ether, thanks to Einstein’s special theory of relativity in 1905.

So it was natural to think of the gravitational field as a field just like Maxwell’s electromagnetic field. It is a field that mediates a force, and it is a field that carries energy and momentum. And whereas the charges that are the sources of the electromagnetic field are electrostatic charges, the “charges” that are the sources of the gravitational field are masses.

Are the masses of all objects involved. It is better to claim: The combined baryonic particles

So far so good, but the gravitational field has another special property that the electromagnetic field does not. Its “charge” is just the inertial mass. The magnitude of the gravitational force acting on a body is proportional to its mass; the magnitude of the inertial force by which the body resists that force is also proportional to its mass. So the body’s mass cancels out of the equations and we’re left with universal gravitation: All bodies accelerate at the same rate in a gravitational field. Which means that when you fall freely in a gravitational field, freefalling bodies near you will either float still or move in a straight line at uniform speed relative to you. In other words, you cannot tell if you’re freely falling or just floating in empty space without an external reference.

This picture is two simple. Consider 4 identical objects in a circle. Each object is attracted, in the same way, by the other 3 objects. Some of the forces cancel out, but the result is that each object is attracted from the opposite direction. If nothing else is involved all the 4 objects will collide at one point. i.e. such a system is instable.

The recognition of this was what Einstein supposedly described as his “happiest thought”, and the consequence was a theory of gravitation that describes gravity using the language of geometry. But does this really mean that gravity is “not a force, just a bending of spacetime?” I think Einstein would disapprove of that statement.

Gravitation should use the language of physics. The problem is that to understand gravity is as difficult to understand that space time is curved. May be curved space time is even more difficult.

In fact I think today he would have even more reasons to disapprove, since we know that we can even describe other forces, like electromagnetism, using similar geometric language. The one difference is that in the case of the electromagnetic force, different bodies respond differently. They see different geometries. Gravity, in turn, is universal: there is only one geometry, and all bodies see that one-and-only geometry.

So there is your answer, then. What is sourced in space by a massive body is the gravitational field, just as a charged body is the source of the electromagnetic field. Both are physical fields containing energy and momentum, and exchanging energy and momentum with other bodies.

Both can be described using the language geometry, but gravity is universal (acts the same way on all bodies), electromagnetism is not.

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