What causes time warping in the space-time?

I was reading through some blogs/articles and watching youtube videos that explained to non-physicists such as myself - how time warping or a gradient in time flow around any object can create gravity. I am able to understand the mechanics (minus the math, I'm not a physicist) of gravity according to this theory, but, a question bugs me:

What creates the time gradient in the first place? Why would the infinitesimally small clocks (or connected particles that move through time at different rates) have the the different tick rates in the first place?

(Kindly be gentle on the math - I am not a mathematician either!)

A follow up question: if an object in space-time is massive enough, such as a black hole, can it stop moving through time... in a way bending time over itself and never letting it go, like light?

• Related: physics.stackexchange.com/q/102910/2451 , physics.stackexchange.com/q/3009/2451 and links therein. Commented Jan 28, 2022 at 9:09
• A nice question (though probably a duplicate), and interesting to see how the answers form a complementary set of distinct viewpoints. Commented Jan 28, 2022 at 11:53
• Your follow-up question should be asked separately, although we already have lots of questions on that topic. Gravitational time dilation is about comparing one clock to a distant one. Locally, you can't avoid moving forward through time at 1 second per second. You might find my answer here helpful. Commented Jan 29, 2022 at 3:22
• It’s not time per se that’s being warped, it’s space-time and it’s caused by mass (energy). Commented Jan 29, 2022 at 20:51
• Thank you all for such amazing answers. Over at stackoverflow such questions are not tolerated; however here at physics community you guys are kinder. I am not sure whose answer to "accept" since each answer has opened my eyes to a hitherto non-existant thought process. Thank you all. Commented Jan 31, 2022 at 14:33

There is a subtle but important distinction you might want to consider which may help you form a better conceptual picture of what happens.

When we talk of time dilation, it is a geometric effect- it means that the length of paths between two points in time can vary. To explain what I mean by analogy, imagine that you and I are standing at a corner of a large square. You walk diagonally across to the opposite corner while I walk around the edge of the square to meet you there. The distances we each walk are quite different because we have followed different paths. If we were carrying pedometers, they would show we had walked different distances, and you would consider that entirely natural- you would not feel you had to make up an explanation about your pedometer being 'pace dilated' relative to mine.

Now, carry that idea over into special relativity and consider time dilation in that context. If you move between two clocks in my frame of reference, and during your journey 4 second pass on your watch, while my clocks show a time difference of 5 seconds between the start and end of your journey, it is not because something has caused your watch to run slow, but that the path through time you have taken between the two points was only 4 seconds long, and your watch has correctly recorded it as such, running at its usual rate to tick off the seconds faithfully.

In general relativity, the mathematics are much more complicated, but the conceptual idea still applies. If you follow one curved path through time its length (in seconds) can be more or less than another curved path through time, so clocks taking the different paths will show different elapsed times. It is not because they are somehow running slow or fast, in the sense of not faithfully recording a true time, but because they are accurately showing real differences in the lengths of their paths through time.

• Wait, does that mean that time is not a dimension, but, a hyper-surface? where multiple paths are possible? I thought paths would be in the space! Commented Jan 28, 2022 at 10:09
• Paths are in both space and time... we cannot help but move through time as we are moving through space. Commented Jan 28, 2022 at 10:17
• Eric's comment is the key point. Space and time form a 4d space, and the path you take between two points can have components in each of 4 dimensions. If you rotate your axes then the magnitude of each component changes, so that the magnitude of the time component might decrease while the magnitude of the spatial components increases. Commented Jan 28, 2022 at 10:58
• @anurag The more you move through space (your speed) the less you move through time. If you add up your velocity through space and time, you get a constant $c$, the speed of light. The sum of velocities through space and time cannot exceed $c$ and it is why you have to sacrifice (give up) your velocity through time in order to travel through space faster, and vice versa. Commented Jan 29, 2022 at 0:14
• I think you are not addressing the question. The answer you wrote is fitting for the following question: "Granted the idea that a source of gravity creates spacetime curvature, how is it that that curved spacetime affects motion of objects traversing it?" However, the actual question does not take spacetime curvature as a given. The actual question is: "What creates the time gradient in the first place?". We have that a physical mechanism for creating the curvature is not known. You shifted (not on purpose of course), to a question that you do know something of an answer to. Commented Jan 29, 2022 at 22:49

The question is "what creates the time gradient in the first place"; "why would clocks ... have different tick rates"?

To answer this we need to remind ourselves that physics never offers explanations in full, but rather it reveals connections between one phenomenon and another, and it shows the structures that lie behind phenomena. So in the present example the structure is that spacetime is a kind of 4-dimensional 'space' with smoothly varying distances and times, and the variation of time and distance is connected to the matter content, and to the way spacetime at one event connects to spacetime at adjacent events. The best we can do to answer the OP's question is say:

"here is the overall framework of general relativity; it has much mathematical elegance and beauty and ties together in one set of ideas a huge number of observed phenomena concerning distance, time, gravity and motion. The idea that spacetime has this nature is such a fundamental concept in general relativity that the theory does not explain it so much as assume it. However, when there is as much mathematical beauty as there is here, we kind of feel as if the assumption is so elegant and captures so much that we feel content that we have understand something profound about the world, even if we don't have a further physical cause to say why it came out like that. Having said that, further physical causes or mechanisms are sought-after in ongoing work on quantum gravity. But they in their turn will have to invoke some different assumptions, so we never fully explain in science, but we do gain a great deal of insight. What we do in science is perceive the nature of the physical world more and more fully and clearly. "

There are a lot of good answers about math on here, but assuming what @anurag is asking is the "philosophical why" (does mass-energy bend space-time), I will point out that we never really ask that question of much of basic physics, but simply accept it as intuitive. This is certainly true of say, F = ma, but even more so of gravity itself. This is not a pedantic point. In the modern world, where it is such an old idea and we all see graphics of planets going around the sun, we take it as intuitive that all masses pull on each other all the time (and across distances). But this idea was very controversial when proposed even to Newton himself.

I guess what I'm saying is, while your question is interesting and should be pursued, I would tell you to ask yourself why you never felt the need to go and ask the internet why Newtonian gravity exists! After all, the idea that everything just pulls on everything else all the time is pretty ridiculous too, it has just become "intuitive" to us.

The reason I am saying this is that, at least for me, getting over the psychological hurdle for some things in more modern physics required me to realize that what I considered "intuitive" was not really intuitive at all, but really just my lived experiences at the scale my senses can measure. So, in a sense, "mass-energy causes space-time to bend" simply because it does. This is not to say that looking for a deeper cause is not fruitful. In fact, it is fair to say that the Einstein gravity "explains" the Newton gravity (shows it to be approximately true in the circumstances observable when it was proposed), by moving the assumption from "all mass pulls on other mass as according to an inverse square of the distance" to "all mass-energy alters the space-time metric as according to the Einstein Field Equations".

What I'm saying is actually the topic of an xkcd. If you don't like that we are still assuming something, in logic you always have to assume something. So perhaps one day someone will explain (basically derive) the EFEs using more basic principles. But it is perfectly valid to take the mass-energy effect on the space-time metric as basically the definition of mass-energy.

Additionally, while it is OK to "abuse the language" a bit, I try to avoid the pop math and pop science terms when understanding things. So I would not recommend getting too hung up on "matter bends space and time" (notice my "all mass-energy alters..." statement above did not use that sort of language). While it can be a helpful analogy, the "bending" of space (and especially time) can get people to think too much about bending objects, e.g. a piece of paper or the "trampoline" in the famous relativity analogy, which I personally think is more confusing than helpful. They then get hung up on "what space bends into" or whatever "bending time" would mean, and of course, "the bowling ball bending the trampoline does not cause the gravity, the gravity causes the bowling ball to bend the trampoline!" (which is why I don't like the analogy).

What really is being said is that "the metric of space-time is a function of mass-energy". Now, this function is complicated enough that when we apply similar metrics to two dimensional surfaces, we would say those surfaces have curvature, and the curvature varies with where the mass-energy is. But when you do differential geometry, Riemannian geometry, or anything like that, the metric does not require anything to "bend into".

FWIW, over any of the usual pop analogies for general relativity, I always found the classic "space-time tells mass-energy where to go, mass-energy tells space-time how to look" to convey the most useful information and understanding.

• I like this answer ... while I try to understand what "spin" is on a fundamental level (or what the heck a muon is), I have no problem "getting" what charge or electrons even though they really are just as mysterious concepts, you just get used to them because you've heard about them since you were little. At the end of the day it's just entities and concepts that have properties and trying to popsci "explain" them is largely pointless.
– eps
Commented Jan 28, 2022 at 21:44
• This answer is quite an eye-opener for me! Commented Jan 31, 2022 at 14:34
• @eps absolutely. Everyone "understands" what charge is in Gen Phys class, not because they actually understand anything, but because they have been rubbing balloons on their hair or their socks on the carpet since they were children. In reality, no one knows what charge is. I actually have a lot of reservations about using the term "spin" because it leads to confusion about a little billiard ball of an electron spinning. Anurag - great! I hope that was the answer to what you were really asking (although the derivation based on least action was interesting). Commented Feb 3, 2022 at 21:13
• Also @eps, if you know what an electron is then good for you. I certainly don't! And really, a muon should not be too much more mysterious than an electron as it is sort of the "same thing" (the -1 charge lepton) but at a larger mass (and much more unstable). Commented Feb 3, 2022 at 21:29

I'm not sure if I understood correctly the question asked by OP, because the answer by Marco Ocram doesn't (in my opinion) answer what I understood of OP's questioning. So feel free to tell me if I am completely wrong with this answer and I will delete this one. I will use some maths but explained maths. If you don't have no time to read the whole thing, please skip to the end " To conclude"

From what I understood, anurag, your question asks for an explanation of why space-time is curved, and especially why time is dilated when a mass is here.

But first of all, I will point out that in general relativity the "gradient in time flow" does not create gravity. It is the energy content, momentum content, pressure content, shear stress content, and flow content of the object that creates gravity. And in a regime where all other effects than time dilation are negligible, time dilation is gravity in this context.

Now, back to your question. There are two possible answers. One saying 'this is the way nature works' and that is not satisfying at all. The other involves a few concepts of analytical mechanics and quantum mechanics à la Feynman:

1. There is something called the least action principle that 'can be justified' in point 2. This principle stated that the action functional must be extremized for the system characterized by it to reproduce classical mechanics. In classical mechanics, action can be thought of as the sum of a quantity called the lagrangian over time. At a given time $$t$$, the lagrangian gives us the difference between the kinetic energy and the potential energy of the studied system at this time. We then write: $$$$\underbrace{S}_{\text{action}}=\underbrace{\int dt}_{\text{sum}} \,\,\underbrace{L}_{\text{Lagrangian}}=\int dt\left[\right. \underbrace{E_k}_{\text{Kin. Energy}}-\underbrace{V}_{\text{Pot. Energy}} \left. \right] \tag{1}$$$$ Concretely the action can be thought of as the accumulation over time of the Lagrangian, that encodes the whole dynamics of the system.

2. In quantum mechanics there is something called 'Feynman path integrals' that is used to find the probability that the studied system goes from a state $$A(t)$$ at time $$t$$ to a state $$B(t+\Delta t)$$ in a given time interval $$t+\Delta t$$. Concretely it is the sum of a phase (that is representative of how much the action has a high value) over all the paths that can be chosen by the system beginning with the stated $$A(t)$$ and ending by the state $$B(t+\Delta t)$$. Here the word 'path' is to be understood as the configuration of the system (if it is a point then the configuration is its position, and if it is the whole space-time then its configuration is its curvature). We write it in different ways but one can find it formulated as follow: $$$$\underbrace{\mathcal{A}(A(t)\rightarrow B(t+\Delta t))}_{\scriptstyle \text{Transition amplitude between} \atop \scriptstyle \text{the states A(t) and B(t+\Delta t)}}=\underbrace{\int_A^B \mathcal{D} x}_{\scriptstyle \text{sum over the paths beginning} \atop \scriptstyle \text{at A and ending at B}} \times \underbrace{e^{i S}}_{\scriptstyle \text{The phase giving how much} \atop \scriptstyle \text{the action has a high value}} \tag{2}$$$$ In fact, the contribution of the paths that satisfy the least action principle is highly dominant in the transition amplitude when this very action has a value that is large in comparison to $$\hbar$$ (the reduced Planck constant, called 'h-bar')

Now time for space-time. The action for the space-time is written as: $$$$S = \underbrace{\frac{1}{2\kappa}}_{\text{Coupling constant}}\times\int dt \underbrace{\iiint d^3 x \underbrace{\sqrt{-g} R}_{\text{Pondered curvature of space-time}}}_{\text{Lagrangian of space-time}} \tag{3}$$$$ The lagrangian for space-time is roughly speaking the integral over the whole space of the curvature of space-time (just as I said before). If we add matter to this action, by simply adding the action of the considered matter, then the least action principle tells us this: $$$$\underbrace{R_{\mu \nu}-\frac{1}{2}g_{\mu \nu} R}_{\text{Curvature of space-time}}=\underbrace{\kappa}_{\text{Coupling to matter}} \times \underbrace{T_{\mu \nu}}_{\text{Matter content of space-time}} \tag{4}$$$$ That is, the known Einstein Field Equation (EFE).

Now, to 'justify' that this relation holds without involving the mysterious least action principle, I will do something really handwavy and I think I've seen this called 'Quantum gravity à la Hawking'. This goes as follow:

• Define the space of configuration of space-time (really hard and unsolved problem)

• Add matter to the system

• Do the same thing as in (2) but with space-time

• Search for the dominant paths contributions and it will match the least action principle provided that the total action of space-time + matter is large in comparison to $$\hbar$$ (at the classical level it is the case).

We write it symbolically like this : $$$$\underbrace{\int \mathcal{D}g}_{\text{Sum of space-time config.}}\times \underbrace{\int \mathcal{D}\Phi}_{\text{Sum of matter config.}}\times \underbrace{e^{iS_\text{space-time}+iS_\text{matter}}}_{\text{Phase associated with space-time + matter}} \tag{5}$$$$ When considering only the configurations' paths beginning with a configuration $$A$$ of the Universe and ending to a configuration $$B$$ of the same Universe, we end up with a transition amplitude between $$A$$ and $$B$$.

To conclude: When time dilation is the only relevant effect of gravity, its cause is the fact that this very time dilation is just the Universe's behaving that contributes the most to the transition amplitude of 'Universe at time $$t$$' and 'Universe at time $$t+\Delta t$$'.

A bit more about 'at time $$t$$' for curved space-time itself:

In General Relativity we describe the curved space-time with coordinates that belong to a flat space-time. So when I say 'the transition amplitude of the Universe from time $$t$$ to time $$t+\Delta t$$', in fact, it is to be understood as the same but with 'for the infinitely far away observer' added.

As I said in the beginning I don't know if this is what you wanted as an answer, so tell me if I am wrong and I will delete this long post.

• First of all, thank you for the detailed yet intuitive explanation. I have a lot of questions regarding the equations, but, I should probably undertake a course in Physics instead of asking them here, but, I cannot help asking: how was EFE formulated? I mean the relation between matter and space-time curvature? I sense a (mutually) recursive definition here - matter causes curvature and hence time dilation through gravity and time dilation then causes gravity. I guess I missed a lot of details there. Commented Jan 28, 2022 at 11:23
• @anurag There's a theorem called the 'Lovelock theorem' that states that the LHS and the RHS of the Einstein equation (4) have the same important property (that is called 'Divergenceless') along with the metric tensor (the '$g$' in the Einstein equation). This means in fact that one can add a term $\Lambda g_{\mu \nu}$ to the equation and the new equation will be also true. This very term is the cosmological constant's contribution to the evolution of the Universe. Commented Jan 28, 2022 at 11:28
• you lost me here... I am not a physicist... could you please be intuitive, as if explaining to a child? Commented Jan 28, 2022 at 11:30
• @anurag Sorry if I've been too technical... Concretely one arrives at the Einstein equation by seeing that the two sides of the equation have the same property. So we equate them. This is not rigorous so this is why I invoked the Lovelock theorem before. Commented Jan 28, 2022 at 11:43
• Lovelock's theorem isn't really relevant for the formulation of the EFE (the theorem is about uniqueness, and many of the assumptions are not intuitive nor historically important for the formulation of GR, hence their much later development). What's important is the dynamics the EFE describe: e.g: they reproduce the dynamics of Newton's laws for weak gravitational fields (correspondence principle), they include the conservation of energy-momentum, and they're non-linear, etc. Commented Jan 28, 2022 at 19:29

Marco's answer is an excellent explanation of time dilation as different path lengths, and I do hope you've read it. But just to supplement it: the reason that paths through time can be "curved" is because matter (and energy, and everything else that exists) bends both space and time.

The common pop-sci picture of a bowling ball on a rubber sheet representing warping of space is very incomplete, because time is warped as well. And in fact in our everyday experience it's the warping of time that matters the most, because our "scale" for time is much more significant. That is, space and time are related by the speed of light. In one second light travels nearly 300,000 km. So on a scale of "natural" units, a second of time roughly corresponds to the distance from the Earth to the Moon.

• I apologize upfront - a whole lot of theoretical knowledge is missing in my head between high-school and now, but, matter bends both space and time - is where I am stuck. Why would matter bend them? Commented Jan 28, 2022 at 11:15
• @anurag General Relativity tells us how much warping is produced by energy, and lets us describe mathematically the exact structure of the warped spacetime. But it doesn't really tell us why energy causes warping. As Jeanbaptiste said, that's just the way the universe behaves. Now maybe there's an even deeper theory that tells us why energy warps spacetime, but then we could ask why that theory operates in the way that it does. ;) Commented Jan 28, 2022 at 11:51
• @anurag Unfortunately, physics does not have an answer as to why matter and energy bends space and time. As of the present, this is considered a philosophical question rather than one that physics can answer. If there is an answer to this, and that might come from quantum gravity (another whole discussion), then it would still lead us to ask more why questions. You can always ask why, and you can keep asking until no one can answer them any more. Commented Jan 29, 2022 at 0:27

While it is the case that physics has achieved numerous successes in explanation of physics taking place, it is not the case that the science of physics is in a position to provide exhaustive explanation.

There is always a 'choose your battles' type of judgement call.

My favorite example: when Newton's Principia was published some contemporary scholars objected along the following lines: "This theory does not offer an explanation of how this Universal Gravity is supposed to act over the vast distances of interplanetary space."

For comparison:
Descartes had offered the supposition that the planets of the solar system are subject to a pushing force from vortices that swirl just outside the perimeter of the orbit of the planet. That is a kind of supposition that arises from a desire to account for observed motion (and change of motion) in terms of contact forces, such as the forces that cause the motion of billiard balls as they move and bump around on a billiard table.

(Newton had as a starting assumption that in celestial motion there must be conservation of momentum, and Newton recognized that any vortex type of theory will always fail to satisfy conservation of momentum.)

Newton acknowledged that he had no hypothesis for how the force of gravity might be transmitted. Newton insisted: this theory so succesful, that in itself is evidence enough that the law of Universal Gravity is correct.

If Newton would have imposed on himself the demand that he should first be able to explain how the force of gravity transmitted then he would only have bogged himself down hopelessly. Newton took the supposition of Universal Gravity as is, and showed that with Universal Gravity terrestrial motion and celestial motion can be described with one and the same law of gravity: Universal Gravity.

That is what I mean with: 'choose your battles'.

It is necessary for a physicist to push hard, that is the only way to expand the body of knowledge. At the same time: avoid bogging yourself down by attacking a problem that is beyond the means available to you.

General Relativity

Specifically to your question: "What creates the time gradient in the first place?"

In the case of GR: the curvature of spacetime (as described by the theory) is a supposition that must be granted in order to formulate the theory at all. The justification for that supposition comes from the success of the theory in accurately describing the physics taking place.

I assert that is the only justification that we have, and I assert that is a totally adequate justification.

Of course, every physicist is keen to move the understanding of physics taking place to a deeper level than the current level, if such a deeper level exists.

At present the means to address the question: "What creates the time gradient in the first place?" are not available.

My personal preference is to think of the spacetime curvature as mediator of gravitational interaction, in the sense that a source of gravitational interaction will induce spacetime curvature around it, and the motion of inertial mass moving through that volume of spacetime is affected by it.

• another eye-opener for me! Commented Jan 31, 2022 at 14:35

Let me give you a little bit different view. In this (naïve) view, the universe (and all objects in it) try to reach equilibrium in terms of the energy they possess and the way this energy flows in the temporal and spatial dimensions. Nice and all, but what does this mean mathematically?

We say, that according to general relativity, objects in a strong gravitational field slow down in the temporal dimension (relatively), this is known as GR time dilation. Now we happen to live in a universe where this attempted equilibrium of the energy flow in the different dimensions manifests in a mathematical formulation known as the constant magnitude of the 4 velocity vector.

Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass.

https://en.wikipedia.org/wiki/Gravitational_time_dilation

When you put an object like in your example into a strong gravitational field (like here on Earth, initially relatively stationary to Earth), (one of) the effect of gravity is that is slows down the object in the temporal dimension (relatively), and so mathematically the 4 velocity vector's temporal component has to change. Remember, the magnitude of the vector has to stay constant, so the spatial components have to compensate (the object has to start moving in space towards the center of gravity).

in short, the magnitude of the four-velocity for any object is always a fixed constant:

https://en.wikipedia.org/wiki/Four-velocity

So naively thinking, the massive (lot of stress-energy and not mass really) object's gravitational field's energy disturbs (changes) the flow of energy of the smaller object (in your example) in the different dimensions (it slows it down in the temporal dimension), thus causing an imbalance, and this has to be equalized by the object (by moving in the spatial dimension towards the center of gravity).

So it is really not mass, rather the stress-energy of the massive object that "warps" spacetime as in your example, and this object's gravitational field's energy that alters the energy flow in the temporal dimension (that could be compensated by a spatial movement), that we perceive as (relative) time dilation.

• a quick question regarding four-velocity, all objects are supposedly moving in space-time with this velocity that has a constant magnitude of "c"; but, where are all objects moving towards? Should I be asking this as a separate question? Commented Jan 31, 2022 at 14:37
• @anurag yes, this should be a separate question, but I think you are asking about the arrow of time as we perceive it. physics.stackexchange.com/questions/235511/… Commented Jan 31, 2022 at 16:13