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When computing the distance at which the escape velocity is equal to the speed of light, the result is equal to the Schwarzschild radius. Is this just a coincidence, or is there a physical reason why the Newtonian approximation give the correct result?

If this is not a coincidence, then why is it OK to use Newtonian physics here?

As explained in this answer, dimensional analysis indicates that this radius depends on $GM$ and on a velocity squared. However, the fact that the two quantities are exactly equal, rather than there being a constant factor between the two is a pure coincidence.

There are two factors I know of. First, and less important, overcoming static friction requires more force than kinetic friction. This applies to all of the internal parts that have to get going/moving past each other, and probably to the rolling friction of the tires.

The major reason, though, has to do with how cars based on internal combustion engines work. See, an internal combustion engine can only supply torque and power when it's already moving. That's why you need an electric starter motor to get the engine going when you start the car. Now think about if the engine were linked directly to the wheels by gears - that would mean if the car is stopped, the engine isn't running. To get over this problem cars have a clutch inside of them that transmits the torque from the engine to the gear box and drive shaft. When the clutch is fully engaged, all of the torque and power are transmitted. As it is in the process of engaging, though, only part of the power is transmitted. This is especially important when starting from rest because it is that partial engagement that allows the wheels to come up to a speed that the engine can supply torque at without stopping.

So, bottom line, the engine needs to be able to supply more torque at low rotation rates in order to get the car moving because not all of the power is being transmitted to the drive train by the clutch.

With an electric motor this is not a problem - they can supply 100% of their torque even at zero rotation.

## Answers and Replies

We just happen to live in a universe where Newtonian physics is not exact. It is perfectly possible to imagine a world where the laws are exact at all speeds (particle physics and some other fields would get problems , but let's ignore the microscopic part here), but experiments show we do not live in such a world.

I'm not too experienced with relativity so I'm not sure, but isn't the 4-force parallel to the 4-acceleration (unless the rest mass is changing)?

The velocity-dependent scalar mass

There is no need for some sort of "vector mass".

As I said in post number 7, if expressed in terms of the 4-force and 4-acceleration, Newton's second law is recovered intact.

Newton made a few assumptions about nature that turned out to be incorrect. For example, Newton's conception of time in the definitions given in the *Principia* as a quantity that moves forward independently without regard to motion (I'm paraphrasing here) was questioned later by Mach who influenced Einstein. Einstein also knew that Maxwell's equations predicted electromagnetic waves that all traveled with the same speed. but relative to what? After the rejection of the *lumineferous ether* largely due to the Michelson-Morley experiment Einstein proposed the two postulates of special relativity - one of which is that the speed of light is the same for all observers regardless of their state of motion. One of the consequences of this postulate (which does not suppose that time runs the same for everyone) is that the amount of time elapsed depends on an observers state of motion. This (and other consequences) of special relativity are only important when the speeds of objects approach the speed of light. If the speeds are low then the predictions made by SR reduce to Newtonian mechanics.

So. to answer the question. It is the axiomatic assumptions that are 'causing the problems'

As others have said, Newton's laws are still applicable in SR if you change the definition of force and momentum to be their four-vector definitions. However, in my limited experience with SR I've noticed that the concept of force (in the Newtonian sense) is not very convenient simply because of how messy this would get when applying the Lorentz transformations. The form of the laws look the same when using four-vectors (which is probably one of the reasons that four-momentum was defined the way it was!), but I would argue that this isn't really *Newton's* laws anymore.

## The bending of starlight is twice the Newtonian prediction

The equivalence principle only applies locally, so local measurements made by a hovering observer will show what you expect. However, the deflection of star light is not a local effect - the light has travelled in from infinity, passed near the Sun, and climbed back up to Earth. So you should not expect equivalence principle based arguments to work.

The simplifications you make to the Einstein field equations in order to recover Newtonian gravity involve neglecting what you might call spatial curvature, keeping only the time-time equation. So, roughly speaking, the solution is that local measurements will measure a tiny "Newtonian" deflection, but if you simply add them together you neglect the spatial curvature. A line of little labs along the light path won't quite fit together the way Euclid says they would - and that failure to fit together is the extra half of the deflection.

The equivalence principle only applies locally, so local measurements made by a hovering observer will show what you expect. However, the deflection of star light is not a local effect - the light has travelled in from infinity, passed near the Sun, and climbed back up to Earth. So you should not expect equivalence principle based arguments to work.

The simplifications you make to the Einstein field equations in order to recover Newtonian gravity involve neglecting what you might call spatial curvature, keeping only the time-time equation. So, roughly speaking, the solution is that local measurements will measure a tiny "Newtonian" deflection, but if you simply add them together you neglect the spatial curvature. A line of little labs along the light path won't quite fit together the way Euclid says they would - and that failure to fit together is the extra half of the deflection.

I could find in the open text

https://web.mit.edu/6.055/old/S2009/notes/bending-of-light.pdf

"Newton’s theory is the limit of general relativity that considers only time curvature general relativity itself also calculates the space curvature." You may get more detail by reading it.

[edit]

As one approaches to sun

time pass slowly

periphery/radius becomes longer

These double effects matter.

I'm not sure what metric you have in mind for the uniform case.

**Summary::** Why does GR predict starlight bending twice Newtonian?

Does this mean that the bending of light in a uniform gravitational field is 2g/c in apparent contradiction of the Principle of Equivalence?

No. In a uniform gravitational field the bending is the usual Newtonian value you mention. The equivalence principle holds throughout a uniform field.

However, the gravitational field of the sun is non-uniform. That is “curvature” in GR. The equivalence principle only holds locally near the sun, over regions small enough to consider uniform. Not over the whole trajectory. The factor of two accounts for the non-uniformity over the whole trajectory.

No. In a uniform gravitational field the bending is the usual Newtonian value you mention. The equivalence principle holds throughout a uniform field.

However, the gravitational field of the sun is non-uniform. That is “curvature” in GR. The equivalence principle only holds locally near the sun, over regions small enough to consider uniform. Not over the whole trajectory. The factor of two accounts for the non-uniformity over the whole trajectory.

Curvature. Many people think that in GR all gravitational effects are due to curvature. But in actuality curvature is specifically the effects of tidal gravity or non uniform gravity.

But you can do a further approximation, setting ##2M/r simeq 2 M/R= ext

Mathematically, you still have both deviations from Minkowski in the metric, i.e., in both components ##g_<00>##, ##g_<11>##. I've never done the calculation for light bending in this approximation but isn't there the same factor of 2 for the deflection angle compared to the naive approximation ##g_<11>=-1##?

The calculation is non-trivial. It's in Hartle's book, culminating in the equation $delta phi = frac<4GM>

We'd need to do the equivalent calculation for a particle in a Newtonian inverse-square force, travelling at the speed of light. I don't think I've got that in my notes anywhere!

PS The classical inverse square deflection can be found here, equation (4.20):

$delta phi = 2 an^<-1>(frac

$delta phi = frac<2GM>

Note also that these are approximations for the particular case of a small deflection. The maths, as it often does, gets messy.

PS The classical calculation is also here (although he just quotes the GR result).

PS The classical calculation is also here (although he just quotes the GR result).

Non-mathematical physics is a dead-end. You might be able to persuade yourself of this, that or the other, but you have no sure way to know where your reasoning will lead you astray.

Once you have a mathematical basis, there is everything to be said.

In any case, the notes I have linked to *are* undergraduate physics. Physics depends on mathematics and not woolly thinking. Phrases like "spatial distortions", "rotating gravitational field", "dragging light" are all so imprecise as to be meaningless.

Non-mathematical physics is a dead-end. You might be able to persuade yourself of this, that or the other, but you have no sure way to know where your reasoning will lead you astray.

Once you have a mathematical basis, there is everything to be said.

In any case, the notes I have linked to *are* undergraduate physics. Physics depends on mathematics and not woolly thinking. Phrases like "spatial distortions", "rotating gravitational field", "dragging light" are all so imprecise as to be meaningless.

It's better not to think of light has bending, but rather moving in a straight line through curved spacetime.

Imagine that you are standing at the Earth's equator and start walking in s straight line due north. I do the same thing at the same time, except that my starting point is a few meters to your left. You will find that although our paths were initially parallel, my path is gradually turning towards you so that we collide at the north pole. You can explain this either by saying that you are walking in a straight line while my path is bending or by saying that we are both moving in a straight line but the surface of the earth is curved so that initially parallel paths draw closer and intersect.

Someone walking in a straight line on a curved two-dimensional space (the surface of the earth) is not a great analogy for a flash of light moving in a straight line through a curved four-dimensional spacetime, but it's as good of an explanation as we can do without the math.

This one is trickier than it seems, because in GR the notion of "gravitational field" is very limited.

Something analogous to a classical gravitational field is only found in a few (important and frequently discussed - the spacetime around a massive body like the sun is one) situations, and none of these involve the sort of rotation that you're considering. For more general cases, in principle we specify the distribution of stress and energy, insert this in the Einstein field equations (64 coupled non-linear partial differential equations), solve for the metric tensor, and then calculate the paths of things (objects in free fall, flashes of light) as they move through the spacetime described by that metric tensor - a "gravitational field" that we consider to be exerting a force on objects just isn't anywhere in the solution.

Flamm's paraboloid is applicable to only one particular situation: a non-moving isolated massive spherically symmetric object like the sun. It won't be much help trying to visualize anything else.

## Follow-Up #3: What is the motion of a falling object in the moon?

Similar to that here on Earth except that the force of gravity is only one sixth as strong as ours. You may remember seeing film clips of Neil Armstrong bouncing around on the Moon's surface.

The reason for the Moon's weaker gravity is that the force due to gravity at the surface of a spherical object is proportional to the mass of the object times G, Newton's constant, and inversely proportional to the square of the radius of the object. Putting it all together you get one sixth of Earth's gravitational force.

## How Stephen Hawking's Black Hole Discoveries Rewrote Physics of Space-Time

Mathematical physicist and cosmologist Stephen Hawking was best known for his work exploring the relationship between black holes and quantum physics. A black hole is the remnant of a dying supermassive star that's fallen into itself these remnants contract to such a small size that gravity is so strong even light cannot escape from them. Black holes loom large in the popular imagination&mdashschoolchildren ponder why the whole universe doesn't collapse into one. But Hawking's careful theoretical work filled in some of the holes in physicists' knowledge about black holes.

### Why do black holes exist?

The short answer is: Because gravity exists, and the speed of light is not infinite.

Imagine you stand on Earth's surface, and fire a bullet into the air at an angle. Your standard bullet will come back down, someplace farther away. Suppose you have a very powerful rifle. Then you may be able to shoot the bullet at such a speed that, rather than coming down far away, it will instead "miss" the Earth. Continually falling, and continually missing the surface, the bullet will actually be in an orbit around Earth. If your rifle is even stronger, the bullet may be so fast that it leaves Earth's gravity altogether. This is essentially what happens when we send rockets to Mars, for example.

Now imagine that gravity is much, much stronger. No rifle could accelerate bullets enough to leave that planet, so instead you decide to shoot light. While photons (the particles of light) do not have mass, they are still influenced by gravity, bending their path just as a bullet's trajectory is bent by gravity. Even the heaviest of planets won't have gravity strong enough to bend the photon's path enough to prevent it from escaping.

But black holes are not like planets or stars, they are the remnants of stars, packed into the smallest of spheres, say, just a few kilometers in radius. Imagine you could stand on the surface of a black hole, armed with your ray gun. You shoot upwards at an angle and notice that the light ray instead curves, comes down and misses the surface! Now the ray is in an "orbit" around the black hole, at a distance roughly what cosmologists call the Schwarzschild radius, the "point of no return."

Thus, as not even light can escape from where you stand, the object you inhabit (if you could) would look completely black to someone looking at it from far away: a black hole.

### But Hawking discovered that black holes aren't completely black?

My previous description of black holes used the language of classical physics&mdashbasically, Newton's theory applied to light. But the laws of physics are actually more complicated because the universe is more complicated.

In classical physics, the word "vacuum" means the total and complete absence of any form of matter or radiation. But in quantum physics, the vacuum is much more interesting, in particular when it is near a black hole. Rather than being empty, the vacuum is teeming with particle-antiparticle pairs that are created fleetingly by the vacuum's energy, but must annihilate each other shortly thereafter and return their energy to the vacuum.

You will find all kinds of particle-antiparticle pairs produced, but the heavier ones occur much more rarely. It's easiest to produce photon pairs because they have no mass. The photons must always be produced in pairs so they're moving away from each other and don't violate the law of momentum conservation.

Now imagine that a pair is created just at that distance from the center of the black hole where the "last light ray" is circulating: the Schwarzschild radius. This distance could be far from the surface or close, depending on how much mass the black hole has. And imagine that the photon pair is created so that one of the two is pointing inward&mdashtoward you, at the center of the black hole, holding your ray gun. The other photon is pointing outward. (By the way, you'd likely be crushed by gravity if you tried this maneuver, but let's assume you're superhuman.)

Now there's a problem: The one photon that moved inside the black hole cannot come back out, because it's already moving at the speed of light. The photon pair cannot annihilate each other again and pay back their energy to the vacuum that surrounds the black hole. But somebody must pay the piper and this will have to be the black hole itself. After it has welcomed the photon into its land of no return, the black hole must return some of its mass back to the universe: the exact same amount of mass as the energy the pair of photons "borrowed," according to Einstein's famous equality E=mc².

This is essentially what Hawking showed mathematically. The photon that is leaving the black hole horizon will make it look as if the black hole had a faint glow: the Hawking radiation named after him. At the same time he reasoned that if this happens a lot, for a long time, the black hole might lose so much mass that it could disappear altogether (or more precisely, become visible again).

### Do black holes make information disappear forever?

Short answer: No, that would be against the law.

Many physicists began worrying about this question shortly after Hawking's discovery of the glow. The concern is this: The fundamental laws of physics guarantee that every process that happens "forward in time," can also happen "backwards in time."

This seems counter to our intuition, where a melon that splattered on the floor would never magically reassemble itself. But what happens to big objects like melons is really dictated by the laws of statistics. For the melon to reassemble itself, many gazillions of atomic particles would have to do the same thing backwards, and the likelihood of that is essentially zero. But for a single particle this is no problem at all. So for atomic things, everything you observe forwards could just as likely occur backwards.

Now imagine that you shoot one of two photons into the black hole. They only differ by a marker that we can measure, but that does not affect the energy of the photon (this is called a "polarization"). Let's call these "left photons" or "right photons." After the left or right photon crosses the horizon, the black hole changes (it now has more energy), but it changes in the same way whether the left or right photon was absorbed.

Two different histories now have become one future, and such a future cannot be reversed: How would the laws of physics know which of the two pasts to choose? Left or right? That is the violation of time-reversal invariance. The law requires that every past must have exactly one future, and every future exactly one past.

Some physicists thought that maybe the Hawking radiation carries an imprint of left/right so as to give an outside observer a hint at what the past was, but no. The Hawking radiation comes from that flickering vacuum surrounding the black hole, and has nothing to do with what you throw in. All seems lost, but not so fast.

In 1917, Albert Einstein showed that matter (even the vacuum next to matter) actually does react to incoming stuff, in a very peculiar way. The vacuum next to that matter is "tickled" to produce a particle-antiparticle pair that looks like an exact copy of what just came in. In a very real sense, the incoming particle stimulates the matter to create a pair of copies of itself&mdashactually a copy and an anti-copy. Remember, random pairs of particle and antiparticle are created in the vacuum all the time, but the tickled-pairs are not random at all: They look just like the tickler.

This copy process is known as the "stimulated emission" effect and is at the origin of all lasers. The Hawking glow of black holes, on the other hand, is just what Einstein called the "spontaneous emission" effect, taking place near a black hole.

Now imagine that the tickling creates this copy, so that the left photon tickles a left photon pair, and a right photon gives a right photon pair. Since one partner of the tickled pairs must stay outside the black hole (again from momentum conservation), that particle creates the "memory" that is needed so that information is preserved: One past has only one future, time can be reversed, and the laws of physics are safe.

In a cosmic accident, Hawking died on Einstein's birthday, whose theory of light&mdashit just so happens&mdashsaves Hawking's theory of black holes.

Christoph Adami, Professor of Physics and Astronomy & Professor of Microbiology and Molecular Genetics, *Michigan State University*

## Why does Newtonian physics fail to cover what quantum mechanics does?

Quantum mechanics deal with matter at the subatomic level, and I was wondering why Newtonian physics do not apply in this situation.

This is a pretty loaded question, that to fully answer requires many fields of physics. However, I will try to give a pretty straight forward answer for you.

So, Newton derived his laws by observation, mostly. Without the modern tools we have now, all Newton could observe was large, macroscopic things. So from his observations, he figured out laws that described how large, macroscopic things behave.

Now, I'm going to switch topics a little bit, but I promise I'll tie it all together (or at least try to!). So, something we have learned about physics is that all possible outcomes are equally likely to occur. We'll use flipping coins for example. Say I flip a coin four times. I am just as likely to get the sequence H-T-T-H as H-T-H-T, or T-H-H-T or really, any other combo, including ones that we think of as "unlikely" like H-H-H-H or T-T-T-T. But why are we more likely to get, say, two heads and two tails when flipping a coin 4 times than all four heads? Because there is only "one way" to get all four heads, that is getting H-H-H-H. However, there are 6 ways to get two heads and two tails, for instance T-T-H-H, H-H-T-T, H-T-H-T, etc. Thus, we are six times more likely to get two heads and two tails as we are to get all four heads. "Where is this going?" you might ask. Well, now imagine instead of flipping coins we are talking about particles of air in a box. Each particle could be anywhere in the box. thus it is possible that all of the air molecules could be jammed into one corner. But why does that never happen? Because statistically, there are many more ways to arrange the air molecules throughout the entire box than there is to arrange the air molecules all in one corner. And when you are talking about the millions of air molecules in the box, the odds are just two small that at any time all of the molecules will be in one corner. Thus we have a law that says "whenever there are large numbers of particles, the most likely combination is the one that will happen." Or using the coin flip again. When only dealing with 4 coins, it is possible that you'll flip all four heads. But if you are flipping a coin 10 million times, never will you ever flip all 10 million times heads. In fact, you will flip very close to 5 million heads each time. No magic here, just statistics.

So, quantum mechanics talks about what happens to single atoms, or small groups of atoms. We were able to discover this with more sophisticated tools that Newton didn't have. And quantum mechanics does show that some really weird things can happen with single particles. But you can think of those really weird things like only flipping 2 coins. Flipping a coin 2x, you would expect to get 1 heads and 1 tails, but you wouldn't be shocked if you flipped both heads, right? Because with small numbers of things, statistically unlikely things can happen. But an interesting thing happens in quantum. You can take the expectation value of any quantum equation (expectation value is just the same as saying "the most likely thing to happen, statistically") and you'll get an equation that agrees with Newton's laws. And if you get enough atoms in one place, while each atom will maybe do something weird, on average they will take on their expectation value.

So to summarize, Newton's equations are really just the expectation values of quantum equations. They are not in conflict, it is just Newton only discovered the versions of the equations that work for large numbers of atoms. This makes sense historically because Newton lacked the tools to see how individual atoms behaved.

## ELI5:Why do Newtonian physics break down at a quantum level?

It's not so much that they break down, rather it's that Newtonian physics is an approximation of how the world works that is not totally correct, but in many cases is accurate enough to be incredibly useful. In such circumstances (like the ordinary motion of a baseball), the inaccuracy is so low as to be practically imperceptible, though it is still there. When things become very small, very large, or very fast, however, the Newtonian model is very inaccurate.

approximation of how the world works that is not totally correct

Does this mean F=Ma doesn't work any more or something? Or one of the other laws?

So really Newton's second law F=ma isn't a law at all?

Serious question and I'm way out of my league so bear with me here:

When discussing the possibility of non-traditional space travel, can people use known physics to understand something that might have a whole new set of laws?

Is there a stubbornness to accept that there might be new/different laws of physics, or just a acknowledged void of understanding being studied eagerly?

What about Relativity and QM?

A lot of people here, I feel, aren't really answering your question as to **why** it breaks down. The easiest way to explain this is with examples. One of the initial ways we discovered quantum mechanics is through exploration of the atom. Initially, we formulated the idea of an atom that was rigid, a building block for molecules, that was built off of newtonian forces. As we discovered more and more about the atom, this explanation became increasingly less consistent with the actual data. For example, we discovered that atoms weren't solid objects, but actually mostly empty space, held together by forces that weren't described in newtonian physics (the strong and weak forces). From there, we discovered principles that led much of the structure of the atom to be based on probability (the probability that an electron was in a certain location) rather than rigid orbits, which would have been more consistent (although still not that consistent) with Newtonian mechanics.

Some other examples include the duality of particles (their ability to be both particles and waves) and quantum tunneling (this duality allows some low-mass particles to pass through solid objects!). The current standard model has all forces mediated by particles, which would have never been even dreamed of by Newton.

Relativity does the same, but breaks down ideas mainly about motion and frame of reference (or in the case of GR, gravity).

So let me see if I got this:

With quantum mechanics, it's hard to know where things are, and sometimes they don't even have mass. So they don't fit into Newton's equations.

As we zoom out the probability of knowing where something is becomes obvious and so we have something that works with Newton's equations.

Another core difference between quantum mechanics and Newtonian mechanics is the Heisenberg Uncertainty Principle. Newtonian mechanics assumes that the precision of measurements of position or momentum of particles are limited only by the accuracy of the measuring devices. In fact, and as QM provides, position and momentum are related so that when one is narrowly constrained (by measurement or interaction), the value of the other becomes indeterminate. The total uncertainty is related to Planck's constant, so it is very important at atomic scales and insignificant for macro objects.

As to why qualities like position/momentum and energy/time are linked by uncertainty relations, it arises from the wavelike behaviour of particles at small scales. A wave may have a continuous height, like a wave at the beach, but it will then appear to be spread over a wide area. When a number of waves interact, you might get a more localised height, but the narrower you want to make that localised wave packet, the wider the range of wavelengths you need. Confining one quality necessarily requires freeing up the other.

Edit: For a short phrase that contains a lot about QM: "The number of ways something could happen affects what does happen".

Theories break down when the postulates behind them are no longer valid. For example, Newton's laws postulate objects at rest stay at rest unless acted upon. At a quantum level, we know this isn't true, because particles are really just smeared out probability distributions telling us we could find the particle in lots of places. This is telling us that firstly the idea of a solid particle is not really accurate at the quantum level, and the idea of something being stationary isn't really applicable too! Newtonian mechanics doesn't take this into account, so when the affects of it become noticeable, it ruins the results of the theory. If we look at the non-quantum limit of systems with quantum mechanics though, we can see Newton's laws emerge as a kind of average behaviour, which is why they work at big scales.

The same thing happens when you go from quantum mechanics (which is a low energy theory) to quantum field theory (high energy). Quantum mechanics postulate a conserved number of particles (to keep wavefunctions normalised or something, I can't really remember), but that clearly isn't very physical because we know particles can be created and destroyed in real life. Quantum field theory can accommodate both all of quantum mechanics and the extra stuff that comes from moving close to the speed of light, like particle production and relativistic effects. If we take a low energy limit of QFT, we get quantum mechanics back!

The standard model itself is what's called an "Effective Field Theory", because it's only valid up to a certain scale. On very high energies, we know it isn't right because it doesn't know about quantum gravity, so we know it isn't going to give us the right answers.

TLDR Theories (like Newton's) work because the things they don't know about don't really make a difference compared to the things they do (like quantum mechanics or relativity). When the things they don't know about start to cause a big effect, they still don't know about it so give wrong answers.

It's like if you try to drive a car without knowing about the steering wheel. It's fine if you're moving on a straight road, because you don't need to know about it. But when you meet a corner, you can't do a good job.

## Follow-Up #8: quantum energies

I think there's one key ingredient here that isn't close to what one would guess based on classical mechanics. It's that the kinetic energy of the electron wave depends on its shape. Specifically, it goes as the second derivative of the wave function with respect to spatial coordinates. That means that the only way to get a small kinetic energy is to have a wave which varies only slowly as a function of position. However, in order to be concentrated in a small region (needed to lower the potential energy) the wave obviously must vary rapidly as a function of position. That's why there is a trade-off.

## Where do Newtonian physics stop and Einsteins' physics start? Why are they not unified?

As a rule of thumb there are three relevant limits which tells you that Newtonian physics is no longer applicable.

If the ratio v/c (where v is the characteristic speed of your system and c is the speed of light) is no longer close to zero, you need special relativity.

If the ratio 2GM/c 2 R (where M is the mass, G the gravitational constant and R the distance) is no longer close to zero, you need general relativity.

If the ratio h/pR (where p is the momentum, h the Planck constant and R the distance) is no longer close to zero, you need quantum mechanics.

Now what constitutes "no longer close to zero" depends on how accurate your measurement tools are. For example in the 19th century is was found that Mercury's precession was not correctly given by Newtonian mechanics. Using the mass of the Sun and distance from Mercury to the Sun gives a ratio of about 10 -8 as being noticeable.

Edit: It's worth pointing out that from these more advanced theories, Newton's laws do "pop back out" when the appropriate limits are taken where we expect Newtonian physics to work. In that way, you can say that Newton isn't *wrong*, but more so incomplete.