According to general relativity planets and the sun bend spacetime, and that is the explanation of gravity. However, does this hold true for smaller objects, like toys, pens, etc.? Do they also bend spacetime?
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 spacetime, and the experiment was able to measure this.
Going to much smaller sizes quantum effects come into play and there are open questions about the behavior of gravity there.
It seems to be the case that all matter and all forms energy will bend spacetime. Whether or not there is some kind of matter we're not aware of that doesn't is another question. But, everything you'd encounter in your day to day life will bend spacetime, and produce gravitational waves.
This indeed holds for small objects. Why shoudn't it? But this will barely affect their motion when they make a close encounter somewhere in outer space. A pen and a lemon will pass each other as if the other wasn't present. When the pen and lemon move on though, their very little influenced trajectories (geodesics) will divert from the trajectories they would have taken in absence of the other. Somewhere far away this difference will become visible. You can even say because of this that special relativity doesn't exist in reality. It's always an approximation (of general relativity). But a good one.
When the objects become very small (elementary particles), a quantum theory of gravity is needed to explain the mutual gravitational behavior of matter on very small scales (Planck scale).
Indeed. You just discovered one of the main building blocks on General Relativity. Anything and everything that does have stress-energy does bend spacetime.
If you use General Relativity instead you'll find that photons make a contribution to the stress energy tensor, and therefore to the curvature of space.
Yes, even massless photons do bend spacetime. Now back to your question, in your example, toys and pens bend spacetime too. The effect is just so small, because here Earth's effect of bending spacetime dominates, and the net effect is what we feel.
Then why is this important? Because in free space, two dust particles will bend spacetime so that they will move towards each other, and will clump into ever bigger pieces. You just realized how galaxies were made.
It is more accurate to say it this way: they affect the determination of (1) which time-like and null trajectories are the trajectories of 0G motions, (2) which space-like paths are straight, (3) which directions in space-time are space-like, time-like or null, (4) which angles are right angles, (5) which space-like directions are directions of "simultaneity" with respect to which time-like directions and (6) which null directions are orthogonal to which other null directions. Items (1-3) are determined by the "connection", items (4-6) are determined by the "metric". The "null" curves are the ones followed by "light-speed" objects.
As best known to date, all of this is the case for all objects of all sizes; but it has not been directly seen - as such - beyond a certain level of smallness, such as by a Cavendish balance. For instance, nobody has successfully probed the gravity of a single atom with a Cavendish balance, much less a single electron or photon.
The 0G trajectories are those which maximize clock time (any non-0G deviation from the trajectory leading to a time dilation) and the straight space-like trajectories are those which minimize path length. The curves are, therefore, called "extremal". There is no similar characterization applicable to null curves that I'm aware of, because both clock time and path length are 0 on null curve trajectories.
In a Riemannian geometry, the connection is determined by the metric (it is the "Levi-Civita" connection of the metric), and the metric is conditioned by the stress tensor of material sources in space-time.
In a Riemann-Cartan geometry (which is the physically relevant geometry, not Riemannian geometry, since only Riemann-Cartan has the ability to directly deal with spinors and fermions), the connection and metric are independent. If the classical 1916 theory of gravity (that is: the theory given by the Einstein-Hilbert Lagrangian and action on a Riemannian geometry) is directly transcribed into Riemann-Cartan geometry, it becomes equivalent to the theory given by the Palatini Lagrangian and action. In it, the connection is not independent of the metric, but is forced (that is constrained) to be the "Levi-Civita" connection of the metric.
If, on the other hand, the Einstein-Hilbert Lagrangian is directly transcribed into Riemann-Cartan geometry ... but with the connection kept independent ... then the theory is that given by the Einstein-Cartan Lagrangian and action. In that case the "contorsion" = (Connection - Levi-Civita Connection) is given uniquely by the "torsion" (each can be expressed as a linear function of the other), and the torsion is conditioned by the spin tensor of material sources. The Levi-Civita part of the connection and metric, are conditioned by the stress-energy tensor, as in the case of Riemannian geometry.
Since torsion does not propagate outside of matter, in either the Palatini/Einstein-Hilbert or Einstein-Cartan cases, then the spin tensor has no effect on gravity outside of matter and the gravity is the same, on the outside, for both cases.
However, since the determination of which trajectories are 0G and which curves are straight is given by the connection, then Einstein-Cartan and Einstein-Hilbert/Palatini will come up with different determinations ... when inside material media, like air, water or solids; the difference being the result of the contorsion. So, in the case of Einstein-Cartan (as also the case generally in Riemann-Cartan geometries), the straight/0G paths ... which are called autoparallel are not the same as the extremal paths.
As explained in the other answers, yes. Its so hard to tell because, compared to the other forces, gravity is very, very, very, very, very, very, very, very, very, very weak. Only when a whole lot of matter gets together does it become significant.
For example, the Earth is 6,000,000,000,000,000,000,000,000 kg of matter, yet we can overcome its gravity with an almost effortless jump. So can a good magnet using electromagnetism.
We don't know why gravity is so weak, it remains a great mystery.
Yes, small objects such as pens, paper, tennis balls etc, all bend spacetime.
Not only does spacetime bend, warp, and stretch, but it also flows. You can see that idea in this video which is a really good visualization of gravity.
Another way to think about gravity is that gravity is a difference in proper time. When an object falls, it falls because there is a gradient difference of proper time. You can see that idea in this video.
I hope that helps.
A large body is an accumulation of many small bodies. Its properties — mass, and with it gravitation and inertia, heat capacity etc. — are nothing but the accumulation of the properties of the constituting small bodies. And vice versa :-).
Remember that all small objects you hold in your hands were once parts of very heavy stars which did visibly bend spacetime. Their atoms got expelled when that star went nova, billions of years ago, and later condensed again to form our solar system. The atoms in your toy bend spacetime just like they did when they were part of that large star.