51

You're using the wording "curved spacetime", but you're still only thinking "curved space" with an independent, linear time. In your curvature model, you're assuming that moving through some 3D spatial point in one spatial 3D direction will experience the same 3D path curvation independent on speed (as if you'd shoot a ball through a ...


49

Let's say space is really a lattice with spacing $\Delta x$. It turns out that this idea has more trouble with experiment than you might think, but we can plow ahead for the purposes of this question. You might propose replacing integrals in physics with discrete sums over individual lattice points, to take a concrete example let's think about the work ...


46

You can always embed a (spacetime) manifold in a sufficiently high-dimensional space (if you have a $d$ dimensional manifold it can be embedded in a space of $2d$ dimensions). But that doesn't specify which space it is - it could be any sufficiently high dimensional space. So assuming it is embedded doesn't tell you anything at all. Hence it is simpler to ...


34

A core idea of special relativity is there is no right frame of reference. It doesn't matter which of the two observers you use as your point of reference, the math will work out either way. Yes, they'll disagree about whose time is what, and they're both right in their own frame of reference. To use a simpler example, let's ignore relativistic effects for a ...


32

Generally when physicists talk about the universe being finite, they are talking about the existence of an upper bound $R$ on the distance between any two points in space. Such an upper bound could arise in several ways - perhaps the universe has an edge - a boundary which cannot be crossed - or perhaps the universe has the topology of a 3-sphere, and so if ...


31

The curvature of spacetime can be separated mathematically into two components, Ricci curvature and Weyl curvature. They are locally independent, but their joint variation over spacetime is constrained by mathematical relations (the second Bianchi identity). General relativity says that the Ricci curvature is determined by the local matter density (stress-...


28

Spacetime curvature makes this possible. Here's an analogy. There are two paths on opposite sides of the equator, at a constant distance from it. Someone walking east along the path north of the equator will have to continually turn slightly left to stay on the path. (If that isn't obvious, imagine it's so far north that it visibly circles the pole.) ...


27

Yes, all forms of matter and energy bend space and time. Size does not come into it, except that where there is more matter and energy gathered together space and time get bent more.


25

There is no working theory that has been, with complete consistency, started from gravitons, and ended up at 4D General relativity in its low energy limit. But the idea of the whole endeavor is the same thing as Electricity and magnetism. You have the classical picture of E&M that is based around electric and magnetic fields, and then you have the ...


24

Black holes are solutions of vacuum EFE on a space-time with singularities. EFE in the vacuum are: $$ G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 0. $$ These are sometimes written as $$ R_{\mu \nu} = 0, $$ which is completely equivalent in spacetimes of all dimensionalities except for $d = 2$. This is easy to see. Write down the contraction of ...


24

I would say the answer is just the scientific principle of parsimony: if an empirically inconsequential entity can be dropped from a theory, it is preferable to drop it. As you have pointed out, the embedding space is not needed mathematically, and thus it also does not impact the empirical claims of the theory. Hence, parsimony tells us to drop it. If you ...


21

Let's approach this by taking a simple analogy. Suppose you and I are in two cars at the equator and we start driving North. Even though we started off driving exactly parallel to each other we will find the distance between us decreases until when we reach the North pole we would collide. Our motion looks like this: (this diagram is taken from my answer to ...


19

You actually are not too far off with your thoughts. There is a subtle issue of terminology. In a system of units the choice of base units is arbitrary. For the SI there are seven base units: second, meter, kilogram, ampere, kelvin, candela, mole. In the SI system all other units are derived from some combination of these and are called defined units. In ...


18

Contrary to popular belief, the HUP is not a principle about the accuracy of a measurement. The HUP is simply a statement that relates the spread of position measurements to the spread of momentum measurements of similarly prepared systems. It is a statistical principle about multiple measurements and their standard deviations; it is not a principle that ...


18

Short answer is: They both do. They both measure the same amount of time dilation. One sees the time slowed for the other observer, and the other observer sees the exact same thing happen to the original observer. Suppose you have an observer A moving at a velocity $v$ relative to another observer B who is stationary (meaning you are seeing things from ...


17

The short answer is no. The components of the Riemann tensor, which is an object measuring the curvature of the manifold at a point, can take any real number value. In classical GR, there's nothing which bounds these numbers, but when they diverge to infinity at a point it indicates a problem with our classical description (see discussions of singularities ...


17

For a given spacetime the metric tensor may be written in both (globally) diagonal and non-diagonal forms depending on what coordinates we choose. For example for flat spacetime one diagonal form (not the only diagonal form) is the Minkowski metric: $$ ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 $$ But we could choose to write the metric using rotating cylindrical ...


17

Recently gravity was measured between two 1 mm gold spheres. (Measurement of Gravitational Coupling between Millimeter-Sized Masses by Westphal et al) Gravity cannot be separated from "bending spacetime". Any force that affects everything equally in a place can alternatively be described as a bent spacetime. So those 90 mg spheres are bending ...


17

The second description arises from attempting to quantise the gravitational field. A full quantisation, so far, has proven elusive. However, we can look at the case of a weak gravitational field, linearise this and quantise. This suggests that the quanta of the gravitational field is a spin-2 particle which is named the gravition. Whilst everyone knows that ...


16

The curvature extends far away from the mass creating it, becoming progressively gentler as the distance increases, and it never completely goes away. The same curvature becomes stronger as the distance decreases, making all the effects of gravity more powerful the closer you get to the mass. These are the characteristics of a field which extends throughout ...


16

You can see this in many different ways. Here is one explanation: The field equations of electrodynamics in harmonic gauge reads \begin{align} \Box A_\mu=J_\mu \end{align} This is a linear theory and the solution can be determined in terms of the sources, i.e. \begin{align} A_\mu(x)=\int d^3x' G(x,x')J_\mu(x') \end{align} where $G$ is the appropriate Green's ...


16

Suppose you work for NASA and are the person who is given the task of announcing the time to the launch of a particular rocket. You will call this time $t=0$, and so five seconds before the launch you will announce "t minus 5..4..3..2..1..ignition". Now this time $t=0$ could have been for example 28th February 12 noon, and everything before that ...


15

Get rid of the planet in your scenario. Just have two objects at the same place and same time in (1+1D) flat spacetime. Let's build our reference frame so that they both start at the origin $(t,x)=(0,0)$, with one moving at $1\,\mathrm{m}/\mathrm{s}$ in the $+x$ direction and one moving at $2\,\mathrm{m}/\mathrm{s}$ in the $+x$ direction. In spacetime, are ...


15

This is a comment, as Andrew's answer is adequate for the problem. I want to point out , which is not clear in your question, the difference between mathematical modeling and the object modeled. When modeling an object mathematically one can use continuous variables by the function of mathematics. If the object modeled has discontinuities, the mathematics ...


15

This question leads us quite quickly to the metaphysical question of what it is that distinguishes physics from pure mathematics. That question is quite subtle so I do not think you will find a complete consensus, among experts, on the answer to your question. To understand physics generally, it becomes more and more useful, as you progress to more subtle ...


15

Even if spacetime is embedded in something bigger, we don't have access to it or any way of making observations of it. This tells you that our theories should only be formulated using quantities that can be computed "intrinsically", without reference to any embedding. So Riemann curvature is in, but mean curvature or second fundamental form are out....


14

The concept of absolute time was perhaps more of a postulate, rather than an axiom. If Newton's physics were based on any axioms, it would have to be the principle of relativity, which is perhaps the most profound concept, fundamental to his laws of motion. This principle was first stated by Galileo$^1$. It states that the laws of physics should be the same ...


13

I think the essential problem lies in the difference between the mathematical meaning of curvature, and the way in which we actually describe a manifold, or a curved space (or spacetime). Although we describe the universe as having spacetime curvature (which is mathematically true), curvature refers to the Riemann curvature tensor, which is a rank-4 tensor, ...


13

The two different formulas are based on different assumptions. The time dilation formula assumes that there are two events in spacetime and that the two events are in the same position in one reference frame. The length contraction formula assumes that there are two worldlines in spacetime and that the two worldlines are at rest in one reference frame. For ...


12

The laws of nature satisfy rotational symmetry about every point (or in other words, the general law of conservation of angular momentum holds everywhere in space). So you can pick any origin you'd like for your coordinate system, and rotate about that origin any way you like, and the laws remain the same.


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