Does a moving object curve space-time as its velocity increases?

Question

We always hear how gravity bends space-time; why shouldn't velocity?

Consider a spaceship traveling through space at a reasonable fraction of the speed of light. If this spaceship, according to special relativity, gains mass as a factor of y as it approaches c, then its gravitational field should increase in strength as well. Hence, space-time should warp.

Note: Changes in space-time, gravity and mass should only be measureable by an outside observer with a different velocity. Those inside of the ship moving with it would not be able to measure the change in these properties.

Answer 1

Start with the gravitational field of the Sun. We are effectively stationary with respect to the Sun, because our relative speed is much less than $c$, and the Sun is rotating at well below relativistic speeds so we expect its gravitational field to be well described by the Schwarzschild metric. And indeed this is true: Newton's law of gravity is the non-relativistic limit of the Schwarzschild metric.

The metric tensor is invariant with respect to coordinate transformations, so if we take some observer moving at near light speed they would also find the gravity round the Sun to be described by the Schwarzschild metric. It will not look the same in the observer's coordinates, that is the individual components $g_{ij}$ will be different, but it will be the same tensor. Since in the observer's frame they are stationary and the Sun is moving, the conclusion is that velocity does not change the spacetime curvature.

Incidentally, this is why a fast moving object does not turn into a black hole.

Answer 2

I assume you're asking whether a moving objects curves spacetime differently than a stationary one as its velocity increases.

Strictly speaking, no: it's the same spacetime geometry either way. The spaceship warps spacetime either way, and all we'd be is talking about it in a different frame. Because of this difference of frames, in a sense the gravitational field is different even though the geometry is the same.

If this spaceship, according to special relativity, gains mass as a factor of y as it approaches c, then its gravitational field should increase in strength as well.

This is not quite right. First, the spaceship does not gain mass. It's true that quantity $\gamma m$ is sometimes called relativistic mass, but this term is redundant with energy, bad at its intended purpose of preserving a superficial resemblance to Newtonian mechanics, and depreciated in physics. In special relativity, mass is invariant: $(mc^2)^2 = E^2 - (pc)^2$ is the same in all inertial frames.

Which is just as well, since the 'gravitational charge' isn't mass, but energy. But it is not a simple proportional increase when we view the spaceship's gravitational field in a frame in which it has a lot of energy.

This shouldn't be surprising if you know a bit of electromagnetism. A moving charge produces an electromagnetic field that has both electric and magnetic parts, since the motion of the charge, i.e. the current, matters. The electric field is enhanced in directions perpendicular to the direction of motion, which we can picture as the initially spherically symmetric field lines getting Lorentz-contracted, thus 'squishing' them close together in the perpendicular directions.

The gravitational analogue of electric current would be momentum, but because gravity is spin-2, stress in addition to energy density and momentum density is relevant to how spacetime is bent. You can see this described in the stress-energy tensor. So the gravitational field is more complicated, but it has an analogous behavior of being strengthened perpendicular to the direction of motion, although its quantitative behavior is different.

In the limit of lightspeed, the electromagnetic field of an electric charge turns into an impulsive plane wave, and the gravitational field of a point-mass behaves analogously, turning into a vacuum impulsive gravitational (pp-)wave.

Answer 3

Looking at the (spatial) velocity of an object alone is like looking at a plane and considering only its north-south velocity and neglecting its east-west motion.

Timelike objects are like cars with only one gear, no brakes, and a gas pedal that is stuck in place. They continue to go forward in time no matter what; all they can change are their direction through spacetime--how much velocity is going forward in time vs. how much is used to travel distances in space. (The steering wheel still works, but that's all, so to speak.)

When you consider both of these velocities together as one quantity, you find that this "four-velocity" of a given object has constant magnitude: the speed of light itself.

Similarly, the "four-momentum" has constant magnitude, proportional to the rest mass of the object.

These are all results from special relativity. General relativity is a bit more nuanced (not in the sense that four-velocity and four-momentum no longer have constant magnitudes--they do--but in how those magnitudes are calculated using the metric).

While different observers will disagree on obviously frame-dependent things like spatial velocity, they will all agree on the magnitudes of vectors in this fashion. You've been told that gravity depends on mass. I'm telling you that it depends on rest mass--which all observers agree on in this way.

Because spacetime boosts are analogous to rotations, boosting a particle and looking at its gravity just results in all spacetime variables (like the metric) being boosted as well. These transformation laws can be tricky for some tensors, but the underlying physical system can still be understood as equivalent to that of a stationary particle. It's just being looked at through another observer's eyes.

Answer 4

Equivalence requires acceleration to curve space the same way gravitating mass does. Hermann Weyl, Zur Gravitationstheorie, Annalen der Physik, 54, 117, (1917) argued that kinetic energy should curve space just as gravity and electromagnetic fields do by entering into the stress energy tensor.

The concept of inertial mass increasing at high speed has been used a lot in science, but is often replaced by non linear terms in the energy momentum equation which has better verification. An argument is made that particle accelerators running at high kinetic energy would not be able to levitate a beam of particles if they had relativistic gravitating mass unless the electric charge was also increasing relativisticly, which is not observed. To conserve the equivalence principle many scientists no longer use the concept of relativistic inertial mass.

In summary an object that is changing speed or direction is expected to generate a gravity wave, usually very small. It is proposed but not proven that an object of high kinetic energy could contribute to space curvature.

Answer 5

When talking about gravity, mass, velocity, energy etc. it’s always good to give the context of the definition i.e. which theory we are discussing. Otherwise things get very confusing.

In General Relativity, gravity is usually said to be a consequence of the curvature (or bending) of space-time. Space-time curvature can be calculated from the total energy-momentum (the stress energy tensor) within a region of space.

In special relativity, the value of the mass/energy and velocity/momentum are relative quantities i.e. they depend on the observer and the measurement of another reference object. The total energy-momentum is invariant i.e. all observers can measure the same value without reference to other objects. The energy-momentum can be decomposed into values for the mass/energy and velocity/momentum but these can be different for different observers (as the observers can only measure them relatively).

For the spaceship in the question, one observer moving with the spaceship would say that the velocity/momentum of the spaceship is 0 and therefore not contributing to the total energy-momentum of the spaceship, another observer moving relative to the spaceship might say that the spaceship has a velocity/momentum which contributes to the total energy-momentum of the spaceship. Both observers will calculate the same value for the total energy-momentum for the spaceship however (as each will also calculate different values for the mass/energy of the spaceship). The same energy-momentum will result in the same space-time curvature. So usually, people don’t say that velocity causes space-time curvature because it is a relative quantity.

It is still an important point that moving objects do result in space-time curvature and therefore gravity – it is not possible for an observer to be at rest relative to all moving objects. This is true even without using General Relativity. A body of material (this could be anything, a lump of metal, a person, a star) is composed of huge numbers of atoms all oscillating in different directions – it is not possible for an observer to be at rest relative to every oscillation. The momentum of each moving component part contributes to the total energy-momentum of the object. In thermodynamics, the hotter the object the faster the component parts oscillate. So an observer at rest relative to the body of material as a whole will measure a higher mass for a hot body than a cold one. In Newton’s gravitational laws the greater the mass the greater the gravitational attraction it causes.