Separating the topics of general and special relativity

So, I have managed to confuse myself beyond the point of repair. I am not a physics student, so my physical knowledge is limited.

Anyway, there are a few topics of relativity, which I can not seem to be able to seperate into either the general or special one. So, for one, I know a lot of things probably overlap, but what is more important to know is which topics do NOT belong into either.

So, when trying to explain special relativity I think I mixed up a couple of things. Correct me if I am wrong:

Special relativity describes that transformation of mass into energy and back is possible. Time is moving relatively (slower for faster moving objects). A ball falling to earth is equivalent to a ball 'falling' in a at g accelerating rocket. A ball in a falling elevator is equivalent to a ball floating in space. Nothing can move faster than light. Light in vacuum is always travelling at the same speed. Light is affected by gravity. Mass alters the time-space-fabric and attracts mass. The actual mass of something depends on the objects absolute velocity. Physical rules are the same in all not-moving systems. Everything moves along spacetime on the shortest possible route.

I know those are in no order what so ever, but are there some things which do not belong in this heap or did I forget anything? I find it hard to find anything that directly separates all those topics.

• Related: physics.stackexchange.com/q/368655 and associated links. May 16, 2019 at 9:19
• "Physical rules are the same in all not-moving systems": movement is relative. The sentence is true if you use intertial systems May 16, 2019 at 10:25
• "Everything moves along spacetime on the shortest possible route" This is not true. Only freely falling objects of very small size move along such routes. May 16, 2019 at 10:32
• "Nothing can move faster than light" this is beyond theory of relativity. First of all, the maximal speed does not need to be speed of light. Theory of relativity only tells you there is such thing as invariant speed (same in all inertial frames) and the value of it. Association with light comes from electromagnetism. Second, the theory itself is not prohibiting objects with superluminal speed, but it does prohibit accelerating above light speed for slower objects and decelerating under light speed for quicker objects. May 16, 2019 at 10:38
• "Mass alters the time-space-fabric and attracts mass" i think this is misleading. 1.) it is stress-energy tensor that curves spacetime not only mass 2.) there is no attraction, only curved spacetime and related concepts (geodesic deviations and such). For attraction tyou must go beyond GR, f.e. to newtonian limit. May 16, 2019 at 10:45

Instead of trying to sort out the details, I'll offer a perspective that can help the details fall into place:

• In general relativity (GR), the geometry of spacetime is dynamic (reacts to its contents) and can therefore be curved.

• In special relativity (SR), the geometry of spacetime is neither dynamic nor curved; it is fixed and flat.

There is also an intermediate subject that doesn't have an accepted name, but it needs a name because we use the subject often. I'll call it "generalized special relativity":

• In generalized special relativity (GSR), the geometry of spacetime is fixed (not dynamic), like it is in special relativity, but it can be curved, like it can be in general relativity.

A fourth combination (the geometry of spacetime is dynamic but flat) does not occur, because if the geometry of spacetime is constrained to be flat, then it's completely specified (except for its global topology), so is has no freedom to react to its contents.

Here are some comments about the importance of each of the three subjects:

• GR is important becasue, of the three models listed here, GR is the best model of the real world. In the real world, influences always go both ways; so if the geometry of spacetime can influence the motion of material objects, then the motion of material object must also be able to influence the geometry of spacetime, like it does in GR. Most importantly, GR includes the fact that matter produces gravity.

• GSR is important because fixing the geometry of spacetime simplifies the mathematics greatly, even if it's curved, and this is sufficient for studying things like the motion of test-objects that wouldn't significantly affect the curvature of spacetime anyway. GSR includes how gravity affects test-objects but doesn't include the fact that matter produces gravity. In GSR, the influence goes only one way. It includes SR as a special case, namely the case when the spacetime curvature is zero.

• SR is important because even if spacetime is curved, the effects of the curvature are negligible in any sufficiently small region of space and time (if no singularities are included). In this sense, SR is a good approximation for the "local" physics in GSR, and therefore also in GR.

• +1 for giving a name to the third category (GSR). It's worth noting that quite a few things usually discussed in a GR course fall under this category (geodesic equations, motion around a black hole etc). May 16, 2019 at 14:01
• Is your segregation of GSR related to the study of QFT on curved manifolds? Or is it an altogether novel version of GR? Just curious.
– user196418
May 16, 2019 at 16:44
• @ggcg It's nothing novel. What I called "GSR" includes the study of QFT on curved manifolds, classical electrodynamics on curved manifolds, and classical test-particle motion on curved manifolds. In other words, it's what's left of GR when we ignore Einstein's field equation and use a prescribed metric instead, but keeping the equations that govern the behavior of everything else (particles and/or fields), whether they're classical or quantum. It's just a name for the part of GR that textbooks typically introduce before the chapter about $R_{ab}-\frac{1}{2}g_{ab}R\propto T_{ab}$. May 16, 2019 at 23:39
• I get it. I just wanted to be sure as I never heard it called that.
– user196418
May 17, 2019 at 2:50

Special relativity addresses the null result of the Michelson-Morley experiment, which failed to measure motion relative to an assumed ether.

I wouldn't say time is moving at all. Rather, measured time and space intervals collected by pairs of observers will only match if their is no relative uniform motion between them. Relative motion causes a mixing of the data of one observer as seen by the other observer. This has some interesting consequences, specifically for simultaneity. However, the bending of light by gravity does not come in to play at all in SR. SR provides us with the correct coordinate invariance of the laws of physics and includes time on equal footing as space. But the space-time of SR is flat and extends to infinity in all directions, like Euclidean space. the notion of boosting mass is a bit out of date. We don't really see things that way any more. The reason is that there is really only one frame of reference that makes sense for measuring a particle's mass, its own rest frame. So M0 is the true mass of any particle. Also, we have a mathematical paradigm for expressing all these results that basically says Energy is the time component of a 4-dim vector, with momentum the other space components. The 4-vector (E, px, py, pz) lives in space-time and the inner product, measure of distance, in this new space give the magnitude of this vector as, -E^2 + |p|^2, I've set c = 1 for simplicity.

The minus sign is very important here. The magnitude is invariant under Lorentz transforms. The "size" of this vector is -m^2 --> E^2 = |p|^2 + m^2. The point is, in this paradigm "mass" is the norm of a vector and is constant.

I know I tossed a lot of math and you said you did not have a background in physics but you can look for some popular books that explain this, including Einstein's text.

Now, once Einstein worked out the correct symmetry for light, and set his postulates for SR, he set out to make all other laws of physics obey the same symmetry. This required modification to Newton's second law of motion and eventually gravity. This is where GR is born. GR replaced the force of gravity with the curvature of the space-time continuum. Astronomical bodies move the way they move because strong sources of curvature create non-trivial geodesic paths. This is why light paths bend in the presence of large massive bodies, the mass curves space-time and the light follows a geodesic. Since light (photons) have no mass their energy-momentum vector is ZERO! That's right those vectors have no length, or null length, in relativity. This is the fundamental separation of GR and SR, the curvature of space and time.

I hope that helps a little.

• So I understood correctly that the curvature of spacetime and the acceleration is GR, but things like the time dialation is SRT? May 16, 2019 at 12:31
• The time dilation and length contraction formulas based on relative velocity between observers and completely SR. That doesn't mean that GR can't produce the same effect. We have GR Doppler effects for example. Gravity creates similar effects by warping space-time. When objects are "moving" on bent space-time we need to account for both effects.
– user196418
May 16, 2019 at 12:52
• But yes, curvature and gravity are nowhere to be found in SR.
– user196418
May 16, 2019 at 12:52

In special relativity you combine the classical momenta $$p=m\bar{v}$$ with the new postulate $$E=mc^2$$ to get a new expression for relativistic momenta $$p=\frac{m\bar{v}}{\sqrt{1-\frac{v^2}{c^2}}}$$. The extra term is known as the Lorentz factor and sets the velocity limit to "c". It will also give that the energy in collisions is higher than classically. When you are dealing with objects accelerated electromagnetically such as for instance in particle accelerators combining this relativistic momenta with the Lorentz Force will give you the acceleration of a charged particle quite perfectly. In special relativity you also have the concept of a time dilation that makes time to slower as you go faster. The slowing down factor is the same as the Lorents factor, when the Lorentz factor due to an objects velocity takes the value of "2" the time experienced by that object slows down with a factor of two.

The new postulate, $$E=mc^2$$ does not really tell you that it is always possible to "transform mass into energy", just that if you lose Energy, you are also going to lose mass and how much you are going to lose.

Also there is of cause the deep underlying question of in relation to what the velocity should be measured.

In general relativity, if we take the most "simple" situation of a spherically symmetric gravitational field, the velocity of light is still the "speed limit". In this case, however, as per a distant observer the velocity of light will vary radially in the gravitational field and become zero at the radial distance that is known as the "Schwarzschild radius".

You also have the added complexity that time experienced locally does not only depend on the "fraction of the velocity of light squared" but also varies with a factor closely relatied to the graviational potential $$\sqrt{1-\frac{2GM}{rc^2}}$$ sometimes called "gravitational time dilation". At the Schwarzshild radius, $$r=\frac{2GM}{c^2}$$, time goes infinitely slow as compared to a distant observer. You also have the related phenomena of "gravitational red shift". The expression for this looks the same as the expression for gravitational time dilation and makes the perceived energy of emitted light for a distant observer go to zero as the source of the radiation moves down towards the Schwarzschild radius.

In general relativity there is also the deep underlying question of the velocity and gravitational potential of the observer which might affect the results radically.

I think you can say that those are the main differences.