This idea is based on a convoluted understanding on space-time. I posted the idea to r/scificoncepts, and it can be found here.
The ideas (as to be determined when put into mathematical form) are either consistent with general relativity (the Einstein equations) and special relativity (speed of light invariance and inertial invariance) or they are not.
The principle I’m working with is how gravity affects time. Generally, the more gravity affecting the spacetime around you, the slower you perceive time (Such as in interstellar where every minute was a hundred years or something.)
This is not an accurate characterization of general relativity. Gravity in General relativity is not separate from space-time but a consequence of space-time geometry. The Einstein equations are complicated to compute for even the simplest of problems but the concept behind them is quite simple. In the most reduced form the Einstein equations can be expressed as (I removed the cosmological constant because it is not relevant to the discussion at hand and/or can be incorporated into the Einstein tensor):
$G_{\mu\nu} = T_{\mu\nu}$
The interpretation is simple: The spacetime geometry (as understood by defining how length is computed between any two points) is determined by the magnitude of the energy-density at any given point. Mass is just one component of energy-density, but becomes the most relevant when discussing things of extreme energy-density like black-holes.
The observed time (or the rate at which things proceed) must vary between different observers ensure that the laws of physics are equivalent in the different frames of reference and that this equation is always observed as accurate.
This is not a difficult concept to envision by analogy. For example take a napkin, lay it flat on a table and draw two points and a line connecting them. Now bend the napkin. When projected onto the table the line will now be shorter than on the napkin. If you move at a constant rate from one point to the other you will go a distance L in time T (v ~ L/T) in the napkin's frame. But in the table's frame you go a distance of $l$ in time t, since $l$ is shorter than L, you will measure a different v if t = T. But if the energy (which is dependent on velocity) must be observed to be the same, then t must not be equal to T. This is NOT what is going on in the Einstein equation: it is far more complicated, however this analogy illustrates the concept.
If more gravity means you perceive time slower than the rest of the universe, it stands to reason that less gravity means you perceive time faster. Currently, you can reach around 0g (but not actually 0g) in the vacuum of space. There could be a workaround with the unknown force of dark energy. Seeing as dark energy counteracts gravity, it could be interpreted as negative gravity. Following the relationship of spacetime, the more negative gravity you have, the faster you will perceive time.
This is based upon a naïve understanding of what is meant by gravity. As stated above, gravity is a consequence of space-time's relationship with energy. Taking away energy therefore leaves an absence of gravitational effects. If you envisioned another force which is opposite gravity (a negative mass perhaps). This negative mass would still add energy to a system, just like two negative charges still adds energy to a system even though they repel each other. To wit, if there existed a particle which repelled matter rather than attracted it, the energy-density would still be positive.
Again this concept can be illustrated by analogy. Darkness for instance is the absence of light. The Candela for instance is a measure of luminous intensity of light, but negative Candela's do not make any physical sense. We cannot have darker than darkness. Or the second, is a measure of time, but we cannot negative seconds. The equations which describe physical laws are just a short hand way of compactly and precisely defining the laws. Making elementary changes to these equations (like adding negative signs) rarely makes physical sense.
Now dark energy is a concept which appears in Cosmology. However dark energy, rather than subtracting from the total energy of the Universe, adds to it so as to account for the hyperinflation of space which cannot occur with only the currently observed amount energy.
The second part of this space-time relationship is time. This part is more tenuous and is potentially a correlation doesn’t equal causation scenario. If you can affect time in any way then you can also affect space. If you condense a lot of time in a certain place (so time is perceived more quickly), then the gravity would decrease. If you slowed down time perception then the gravity would increase.
Space is not separate from Time in General Relativity. Likewise Time is not separate from Space. Spacetime is determined by the distribution of energy; of which mass is a component. The relationship is a complex mathematical expression which cannot be characterized by saying "More gravity == expanded time, reduced time == less gravity". This is not a valid interpretation of the Einstein equations as I detailed above.
(EDIT)
I saw mentioned in the comments to the question the idea of simply choosing a spacetime curvature which could have the desired effect.
The issue here is violating the geodesic equation. Additionally and ultimately the consequence here is the violation of energy conservation. Energy, whether potential or kinetic represents the ability to move objects through space: the higher the energy the faster the motion through space (the higher the energy transferred -> the greater the force -> and the faster the object accelerates).
If an object had a property which produced a net negative energy density; this that would mean that this object would simply remove energy from the system. This clearly violates the conservation of energy.
Now I am aware of the idea of the Alcubierre drive, Quantum fluctuation..etc. This is very different however, since the Alcubierre drive produces a negative energy density, with a corresponding positive energy density elsewhere so the total energy change remains zero.
Additionally, negative energy densities in Casimir effect are only the effects of virtual particle interactions. Off-shell interactions are not interpreted as physical since they cannot be directly observed and are simply the result of the mathematics required to perform calculations in QFT. Simply put, net negative energy densities do not appear in on-shell calculations, nor is General Relativity compatible with QFT rendering conjecture of this nature immaterial.
